Geometry Questions

Introduction :

Geometry is a section in math, which deals with many aspects regarding shapes, figures. they involve with construction, study of their properties, area , volume, etc. They include study of solids too. Geometry deals with the entire concepts related to the shapes, solids, etc. Sample questions about the intersecting lines, area are in the following section.


Example geometry questions:


Here are few example geometry questions:

Geometry question 1:

Find the area of the triangle formed by (5,2), (-9,-3), (-3,-5)

Solution:

The formula for finding the area of the triangle formed by  (x1,y1), (x2,y2), (x3,y3)  is 1/2 | [x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)] |

Applying the formula,we get

1/2 | [5(-3 + 5) -9(-5 -2) -3(2 + 3)]|

1/2 |10 + 63 – 15|

1/2 |58|

Hence the area of the triangle is 29.


Geometry question 2:

Find the point which divides the line segment joining (-1,2) and (4,-5) in the ratio 3:2

The formula for a point which divides the line joining A(x1,y1) and (x2,y2) in the ratio l:m is

`((lx2 + my1)/( l + m) ,(lx2 + my1)/( l + m)) `

Applying the formula,

The point is

`((3 * 4 + 2 * (-1)) /( 3 + 2) , (3 * (-5) + 2 * 2)/(3 + 2))`

`((10)/(5),(-11)/(5))`

Hence the point is (2,-11/5)

Geometry Shapes Definitions

Introduction to Definitions of Geometry Shapes :

A branch of mathematics concerned with the properties of lines, curves and surfaces usually divided into pure algebraic and differential geometry in accordance with the mathematical techniques utilized. Here we have to learn about different geometry shapes definition. Please express your views of this topic Types of Quadrilaterals by commenting on blog.

List of Different geometry shapes whose definition follows:

Triangle

Quadrilateral

Pentagon

hexagon

Heptagon

Octagon

Square

Circle


Definitions of Geometry shapes - Triangle, Quadrilateral, Pentagon ,Heptagon, Octagon:


Triangle:A triangle is defined as a polygon with three vertices and three sides which are line segments. A triangle with vertices X, Y, and Z is denoted triangle XYZ.It is generally classified as there types they are isosceles, equilateral, and scalene triangle.

Quadrilateral:It is a plane figure having four straight sides. The word quadrilateral is made of the words quad represents the four similarly laterals represents the sides. The sum of interior angles equal to 360 degree. There are many types of quadrilateral are there like trapezoid, parallelogram, rectangle, kite.

Pentagon:A 5-sided polygon (a flat shape with straight sides)

Heptagon:A plane figure having seven sides. If all the interior angles of a heptagon are equal then it is known as regular heptagon .It is also called as septagon.

Octagon:An 8-sided polygon (a flat shape with straight sides).

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Definitions of Geometry shapes - Square, Circle, Hexagon:


Square:A 4-sided polygon (a flat shape with straight sides) where all sides have equal length and every angle is a right angle (90°)

Circle:It is a plane curve formed by the set of all points of a given fixed distance from a fixed point .The fixed point is called the centre and the fixed distance the radius of the circle.

Hexagon:A polygon with six sides. A regular hexagon has all its side of equal length and hence all vertical angles are equal and each being 120 degree. A vertex contains 3 diagonals and hence it has fully 9 diagonals.

Geometry Exam Questions

Introduction to Geometry exam questions:

Geometry is a part of mathematics which is related with the questions of size, relative position of figures and with properties of space. Geometry is one of the branches of sciences. A physical arrangement will  show geometric forms or lines.

Geometry analyzes the relations, properties, and the measurement of solids, planes, lines, and angles the science which covers of the properties and relations of magnitude the geometry of the relations of space.

Preparation for the geometry exam questions are essential. Some of the geometry exam questions are given below: Having problem with Exterior Angles Definition keep reading my upcoming posts, i will try to help you.


Geometry Exam Questions:


Find the correct answer to the following geometry exam questions

Q 1: If the circle has diameter of 8, what is the circumference?

A. 6.28

B. 12.56

C. 25.13

D. 50.24

E. 100.48

Answer: A

Q 2: Identify the form of the statement All City College students love math.

a)If a person loves math, then the person is a City College student.

b)If a person is a City College student, then the person loves math.

c)If the person is not a City College student, then the person does not love math.

d)None of the above

Answer: B

Q 3: Name the property of equality that can justifies the statement, If m1+25=180 , then m1=155 .

a) Symmetric

b) Transitive

c) Substitution

d) Subtraction

Answer: D

Q 4: Find the measures of the sides of the equilateral delta PQR if PQ=5x-7 and PR=2x+5.

a)39

b)4

c)12

d)13

Answer: D

Q 5:  All the parallelogram have

a) opposite angles that are supplementary.

b) diagonals that are congruent.

c) four congruent opposite sides that are congruent.

Answer: D


Additional geometry exam questions:


Q 1:  Name the property of the equality that justifies the statement, If m1+25=180 , then m1=155..

a) Symmetric

b) Transitive

c) Substitution

d) Subtraction

Answer:D

Q 2:  What symbol is used to indicate in writing the two line segments are identical?

a)  //       b)=   c) ?       d)?

Answer: D

Q 3: What is the principle basis of the inductive reasoning?

a) Postulate   b) Past Observation     c) Definitions      d) Theorems

Answer: B

Q 4:

Two circles both of radii 6 have exactly one point in common. If A is a point on one circle and B is a point on the other circle, what  is the maximum possible length for the line segment AB?

a) 12   b) 15     c)24    d)20

Answer: C

Geometry Congruence

Definition for geometry congruence:

In Geometry, we study different figure, their properties the relations between them. Every figure has its shape, size and Position. Given two figures you can easily decide whether they are of the same shape.

Figures having same shape and size  and angles are called congruent. Congruent means equal in all respects of the given figure. If two figures are congruent then it means that the size, shape and measurement of the first figure correspond to the size, shape and measurements of the second figure. I like to share this Symbol for Congruence with you all through my article.


Congruence for geometry:


If two persons compare the size shape of their fore-hands, they will do so by comparing thumb with thumb, fore-finger with fore-finger etc. Thus thumb corresponds to thumb. Similarly the two fore-fingers correspond to each other.

When we put a figure on another figure in such a way that the first figure covers the other figure completely i.e. all parts of the first figure completely cover the corresponding parts of the other. Then these figures will be said to be congruent to each other.

The relation property of two figures being congruent is called congruence. When two figures are congruent we denote them symbolically as. One figure ≅ second figure. The property of congruence, as you know is symbolically represented as ≅.


Important points in congruence of geometry:


Congruency in geometry means to be equal in all respect.
Two figures can be said to be congruent only when all parts of one are equal to the corresponding parts of the other.
The property of congruence of figures is called congruency.
If two line segments are congruent then they must have the same length.
If two angles are congruent then their measures must be equal.
Two triangles are congruent only
Two squares are congruent if they have the same side length.
Two rectangles are congruent if they have the same length and breadth.
Two circles are congruent if they have the same radius.
Sum of the three angles of a triangle is equal to 1800 therefore. if the measures of any two of them are given the third can be ascertained.

Coordinate Systems Geometry

Introduction of coordinate systems geometry:

Geometry is one of the basic and oldest topics in the mathematics. Geometry is used to study the characteristics and properties of the figure. Let every point on a straight line is associated with exactly one real number only. Rene Descartes, a mathematician who is the first man to introduce an algebraic geometry of coordinate systems. A plane is a collection of points in a space of the coordinate systems of geometry.Is this topic Lateral Area of a Rectangular Prism hard for you? Watch out for my coming posts.

About the coordinate systems:


Let us consider a sheet of the paper as the plane and draw two fixed perpendicular straight lines in that plane of the paper which will be intersecting at a point.

We always draw a straight line in horizontal direction and the other line will be a vertical line. These two lines which will meet at a common point and it is named as O and called the origin.

We represents that the horizontal as x–axis and the vertical line as y–axis.

The two lines which divides the plane into four parts namely quadrants. These quadrants are named as I quadrant, II quadrant, III quadrant and IV quadrant in geometry systems.

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Constructing co-ordinates system geometry:


Consider any point P in the plane. This point P lies in a quadrant.

From P, draw a straight line parallel to the y–axis to meet the x–axis at the point L, and draw a straight line parallel to the x–axis to meet the y–axis at the point M.

Let 'a' representing the point L on x–axis and 'b' representing the point M on y–axis.

If P lies on the x– axis, then b = 0.If a = 0, then a> 0 and b > 0. If a < 0 and b > 0, then P lies within the II quadrant.

If P lies within the III quadrant, then a< 0 and b < 0. If a > 0 and b < 0, then P lies within the IV quadrant If P is the point O, then a = 0 and b = 0. The number a is called the x–coordinate of the coordinate system of point P and the number b the y–coordinate of the coordinate systems of geometry.

The plane now is called the rectangular coordinate plane systems or the xy–plane.

How To Solve Geometry

Introduction :

Geometry is a branch of mathematics, which deals with lines, curves, solids, surfaces and points in space. In geometry, a point has a position only and is represented by a dot. A point has no length, width, or thickness. A line has length but no thickness or width. The position of a line with end points are called line segment.


How to solve Geometry Problems:


Geometry Problem 1:

Solve the equation of the straight line parallel to 6x + 4y = 12 and which passes through the point (3, − 3).

Solution:
The straight line parallel to 6x + 4y − 12 = 0 is of the form
6x + 4y + k = 0 … (1)
the point (3, − 3) satisfies the equation (1)
Hence 18 − 12 + k = 0 i.e. k = -6
3x + 2y - 6 = 0 is the equation of the required straight line.

Geometry Problem 2:

Solve the equation of the straight-line perpendicular to the straight line 3x + 4y + 28 = 0 and passing through the point (− 1, 4).

Solution:
The equation of any straight- line perpendicular to 3x + 4y + 28 = 0 is of the form4x − 3y + k = 0
the point (− 1, 4) lies on the straight line    4x − 3y + k = 0
− 4 − 12 + k = 0 ⇒ k = 16
the equation of the required straight line is 4x − 3y + 16 = 0

Geometry problem 3:

The lengths of two sides of right triangle are 7 cm and 24cm. Find its hypotenuses.

Solution:

AC = 7 cm
BC = 24 cm
AB  = ?
AB^2 = 7^2 + 24^2
= 49 + 576
AB^2  = 625
AB = √625 = 25

Thus, the hypotenuses are 25 cms in length.



Geometry Problems to practice:


1) Solve the equation of straight line passing through the points (1, 2) and (3, − 4).

Ans: 3x+y = 5

2) Solve the distance between the parallel lines 2x + 3y − 6=0 and 2x + 3y + 7 = 0.

Ans: √13 units

3) Find the point of intersection of the straight lines 5x + 4y − 13 = 0 and 3x + y − 5 = 0.

Ans: The point of intersection is (1, 2)

4) Solve the equation of the curve formed by the set of all those points the sum of whose distances from the points A (4, 0, 0) and B (-4, 0, 0) is 10 units.

Ans: 9x^2+25y^2+25z^2-225=0.

Geometry Expression

Introduction for geometry expression:
Geometry expression is one of the most important lesson in the geometry. Geometry gives the different geometrical shapes and diagrams in our daily life such as articles in the houses, wells, buildings, bridges etc. The word ‘Geometry’ means a learning of properties for diagrams and shapes. The basic shapes of geometry are point, line, square, rectangle, triangle, and circle. The geometry of plane figure is known as Euclidean geometry or plane geometry. Here we are going to learn about examples of geometry expression problems and practice problem. Understanding Definition for Trapezoid is always challenging for me but thanks to all math help websites to help me out.


Example problems for geometry expression:


Problem 1:

Find the equation of the line having slope 1/2 and y-intercept -3.

Given:

m = `1/2` , b = -3

y = mx +b

Solution:

Apply the slope-intercept formula, the equation of the line is

y = `1/2` x + (-3)

2y = x - 6

x - 2y - 6 = 0

Problem 2:

Solve of geometric expressions based on two angles of a triangle measure 35° and 75° and to find the measure of the third angle.

Solution:

Let the measure of third angle be X

We know that the sum, of the angles of a triangle is 180°

35° + 75° + x = 180°

Solving the expression we get,

110° + x = 180°

X  = 180° – 110°

= 70°

Problem 3:

Find the midpoint between the given points (1, 3) and (3, 7).

Solution:

Given: x1 = 1, y1 = 3 and x2 = 3, y2 = 7

Formula:

(xm, ym) = [`(x1 + x2) / 2 ` , `(y1 + y2) / 2` ].

=` (1 + 3) / 2` ,` (3 + 7) / 2`

= `4 / 2` , `10 / 2` .

= 2, 5

Answer:

The midpoint for the given points (2, 5)

Having problem with Arc Length Examples keep reading my upcoming posts, i will try to help you.

Practice problems for geometry expression:


1. Find the area of the rectangle with the length is 13 cm and breadth is 10 cm

Ans: 130

2. Solve the geometric expression based on the triangle ratio. The triangle ratios are 3: 2: 4. Find the angles of a triangle.

Ans: 60°, 40°, 80°

3. Find the slope and y-intercept of the line equation is 3x + 4y + 5 = 0.

Ans: Slope(m) = -3/4, y - intercept(c) = -5/4

Definitions of Geometry

Introduction to definitions of geometry:

"Earth-measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. A mathematician who works in the field of geometry is called a geometer. Source – wikipedia.


Importants definitions of Geometry :

There are various terms and definitions involved in geometry. Some of them are listed below:

Lines:

In geometry if A and B are the two points in the plane. There is only one line AB containing the points. The region where two points connects via the shortest path and continues indefinitely in both the directions is referred as a line.

Line segments:

Line segment is a part of line between two points. The line Segments that intersect at an angle of 90° is called Perpendicular line segments and the line segments that never intersect are known as parallel line segments.

Angles:

An angle is an inclination between two rays with the same initial point.

Right angle:

Angle that measures 90° is referred as right angle

Acute angle:

Angle that measures less than 90° is referred as Acute angle

Obtuse angle:

An angle that measures more than 90° is referred as Obtuse angle.

Scalene triangle:

A triangle in which all three sides has different lengths is known as Scalene Triangles.

Isosceles triangle:

A triangle with two equal length sides and also with two equal internal angles is referred as an isosceles triangle.

Equilateral triangle:

In geometry if a triangle has the equal length on all three sides, then it is referred as Equilateral triangle.

Axioms:

Certain statements are assumed as being true without proof apart from the theorems. Such assumptions are called axioms.

Complementary angles:

Two angles are said to be complementary if their sum is 90° and each is called the complement of the other.

Supplementary angles:

Two angles are said to be supplementary if their sum is 180° and each is called the supplement of the other.


Important definitions of geometry: Circles


The followings are some of the important definitions of geometry in circles.

Circles:

A and B are two concentric circles with radii r and R respectively and O is the center of the circle.

Circumference:

The distance around a circle is called the circumference of a circle.

Radius:

It is the distance from center of a circle to any point on that circle's circumference.

Chord:

Chord is a line segment joining two points on a curve.

Arc:

Part of a curve is referred as an arc.

Concentric circles:

Circles having the same center but different radii are called concentric circles.

Intersecting circles:

Two circles are said to be intersecting when they cut at two different points.

Touching circles:

In geometry two circles are said to touch one another if they meet only at one point. The point at which they touch one another is called the point of contact.

6th Grade Geometry Problems

Introduction:

Sixth grade geometric contains the basic of geometricals .It includes the topic of geometric in Points, Lines ,Line segment, Triangles, Types of triangles, circles, Angles, Types of Angles, Quadrilaterals

Geometric Definitions:

Point: A point   determines the location of particular area.

Line:   A line through two points A and B is written as AB.. It extends

Indefinitely in both directions. So it contains countless number of points. Two points are enough to fix a line

Types of lines:

Intersecting lines
Parallel lines
Perpendicular lines

Triangles in Geometry:


Triangles:

A triangle is a three-sided polygon. In fact, it is the polygon with the least number of sides

Types of Triangle:

Equilateral Triangle
When all the three sides of a triangle are equal to each other, it is called an Equilateral triangle. Each angle measures to 60 degrees. It is a type of regular polygon.

Isosceles Triangle
When two sides of a triangle are equal it is called an Isosceles triangle. It also have two equal angles.

Scalene Triangle
When no two sides of a triangle are equal the triangle is called Scalene triangle. It has three unequal sides.

Area of triangle: 1/2(Base*Height)

Perimeter of Triangle: (Sum of three sides)

Example problem:

1.Find the area of triangle base is 4cm,height is 2cm

Solution:

Area=1/2(4*2)

=8/2

=4cm2

2.Find perimeter of Triangle side lengths are 5cm,5cm,8cm

Solution:

Perimeter=(A+B+C)(Sum of three side lengths)

A=5, B=5, C=8

=(A+B+C)

=5+5+8

=18cm


Angles and Circle in Geometry


Angle:

Right Triangle
. Right angle is equal to 90 degrees. It obeys Pythagoras theorem.

Acute angle
. Acute angle is an angle which is less than 90 degrees.

Obtuse angle
An Obtuse angle is an angle which is greater than 90 degrees but less than 180 degrees.

Acute angle:
Acute and Obtuse triangles are also called as Oblique triangles because they don’t have any angle measuring 90 degrees.

Quadrilateral:
A four sided polygon is a quadrilateral. It has sides and 4 angles

Circle:

Are of circle=Pi*r*r

Circumference of Triangle=2*Pi*r

Diameter=2*Radius

Example:

Find the area  and circumference of the circle when the radius is 4cm?

Solution:

1.     Area=Pi*r*r (r=4) (Pi=3.14 constant)

=3.14*4*4

=50.24cm2

2. Circumference =2*pi*r

=2*3.14*4

=25.12cm

Answers to Geometry Homework

Introduction to answers to geometry homework:

Learning geometry has traditionally been regarded as important in the secondary schools, at least partly because it has been the primary means of teaching the art of reasoning.

Geometry is a theoretical subject, but easy to understand, and it has many real practical applications. Eventually, geometry has evolved into a skillfully arranged and sensibly organized body of knowledge. I like to share this Triangular Prism Net with you all through my article.


Part 1 -answers to geometry homework:


Geometry homework example 1:

If the perimeter of a cube is 52.5 ft, find its surface area.

Geometry homework solution:

Perimeter of a cube P=12a

52.5=12a

a=52.5/12

a=4.375 ft.

So, the value of a=4.375 ft.

Surface area of a cube SA=6a2

=6(4.375) 2

=6(19.14) ft2

=114.84 ft2

Answer of example 1: Surface area of cube   = 114.84 ft2

Geometry homework example 2:

A barrier of length 15 m was to be built across an open ground. The height (h) of the wall is 5 m and thickness of the barrier is 32 cm. If this barrier is to be built up with bricks whose dimensions are 25 cm × 18 cm × 12 cm, how many bricks would be required?

Geometry homework solution:

1 m=100 cm

Here, Length = 15 m = 1500 cm

Thickness = 32 cm

Height = 5 m = 500 cm

Therefore, Volume of the barrier = length × thickness × height

= 1500 × 32 × 500 cm3

Now, each brick is a cuboid with length = 25 cm, breadth = 18 cm and height = 12 cm

So, volume of each brick = length × breadth × height

= 25 × 18 × 12 cm3

So, number of bricks required =volume of the barrier divided by volume of each brick.

Substituting the values,then we get the final answer.

= (1500 × 32 × 500)/ (25 ×18 × 12)

= (24000000)/5400

=4444.44

Answer: The barrier requires 6416 bricks.

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answers to geometry homework:


Geometry homework example 3:

A line passes through (–3, 4) with a slope of -1/5. If another point on this line has coordinates (x, 2), find x.

Geometry homework solution:

Slope m= (y2-y1)/(x2-x1)

-1/5= (2-4)/ (x-(-3))

-1/5= (-2)/(x+3)

We can take cross multiplication.

-1(x+3) =5(-2)

-x-3= -10

-x=-10+3

-x=-7

In both sides cancel for the negative sign and then we get the final answer.

x =7

Answer: The x value is 7

Positive and Negative Angles

Introduction to Positive and Negative Angles:

An ANGLE is strong-minded by rotating a ray about it's endpoint. The initial location of the ray is the INITIAL SIDE of the angle, and the ending position of the ray is its TERMINAL SIDE. The endpoint of ray called the VERTEX.

The unit circle of the angle is said to be in STANDARD POSITION because its vertex is the origin, and its initial side lies on the x-axis. This is also called positive angle, meaning it's created by a COUNTERCLOCKWISE rotation.
The unit circle of angle is in standard position, but it's called a NEGATIVE ANGLE, since it is created by a CLOCKWISE rotation. I like to share this Inscribed Angles with you all through my article.


Rules of Positive and Negative angles:

Positive Angle:

An angle formed by anti-clockwise rotation is a positive angle. In the figure initial side is OX. When these side is rotated by an angle θ in counter clockwise direction then angle is generated is called positive angle.

Negative Angle:

An angle generated by clockwise rotation is a positive angle. In these diagram let the initial side is OX. When these side is rotated by an angle θ in clockwise direction then angle called as negative angle.

Rule I:

Sign of an angle is always positive when measured in anti-clockwise direction.

Rule II:

Sign of an angle is always negative when measured in clockwise direction. Understanding Volume of Right Prism is always challenging for me but thanks to all math help websites to help me out.


Example of Positive and Negative Angles:

Positive and Negative Angles:

Positive Angles start from 0 degrees and turn around counterclockwise.

Negative Angles start from 0 degrees and turn around clockwise.

You can translate your negative angle to its equivalent positive angle by adding 360 degrees to it until it turns positive.

Once it is positive, you can pleasure it the same as you would any other positive angle in the quadrant that it is in.

Example 1:

Angle is -135 degrees.

sum  360 degrees to it until it turns positive.

It turn positive then first time we add 360 degrees to it.

The equivalent is positive angle is 225 degrees.

It is in the quadrant of 3.

Example 2:

Is 300 not same as -300?

Solution:

The answer to this question is NO. Here why the angle have  two attributes attached to it: Degree of rotation (or magnitude of rotation) and Direction of rotation (clockwise or anticlockwise). While those wo angles have same degree of rotation, direction of rotation is just opposite as signified by there opposite signs. Therefore those two angles are different.

Angle of Rotational Symmetry

Introduction to angle of rotational symmetry:

Rotational symmetry is definite as angle, while we rotate as alternate a shape in its center point; you may notice that at a certain angle, the shape coincides with its not rotated itself. When this happens, the shape is said that it have rotational symmetry. A shape has rotational symmetry if it fits on to itself two or more times in one turn. The number of times the rotational symmetry is the shape fits on to itself in one turn. Is this topic Straight Angle Definition hard for you? Watch out for my coming posts.


Type of symmetry:

Symmetry has,

Line symmetry
2D rotational symmetry
3D rotational symmetry
A 2D shape has a line of symmetry if the lines separate the shape into two share equally – one being the mirror image of the other.

Rotation: whatever rotate the shape on it around, every rotation has depending on center point and an angle.

Translation:  Translation is to be in motion without rotating or reflecting, every translation has depending on distance and

a direction.

Reflection: reflection is seemed mirror image. Every reflection has a mirror line.

Glide Reflection: With the direction of the reflection line, glide reflection is the symmetry of its collection of reflection and  translation.

Angle of rotation:


When you can turn a figure around a center point by less than 270° and the figure appear that there is no changed, and then the figure has rotation symmetry. The middle of rotation, and then the smallest angle require turning and the point around which you rotate is called the angle of rotation.

For instance take any figure on the left can be turned by 160 degree the same way you would turn an hourglass and it look like the same. Her take 2nd figure one can be turned by 120 degree and other one is 72 degree. The 2nd figure comes from 72 degree that it has five points, just rotate it until it looks the same; we need to make 1 / 5 of a completed 360 degree. So 1 / 5 * 360 degree = 72 degree.

Geometry Book Answers

Introduction:

Geometry is nothing but the act of determining the dimensions or volume of an object. Buildings, cars, planes, maps are of great examples of geometry. The plane geometry, normally used in finding the area, perimeter, circumference of  Two-dimensional figures like triangle, circle, rectangle, rhombus, trapezoid, quadrilateral etc.  Let us solve some sample problems from plane geometry. Understanding Geometry Lines is always challenging for me but thanks to all math help websites to help me out.


Sample Geometry Problems:


Example 1:
The radius of a right circular cylinder is 7 cm and its height is 20 cm. Find its curved surface area and total surface area.

Solution:
Radius(r) = 7cm, Height(h) = 20 cm.


Curved surface area = (2`pi`r h )= 2 * 22 / 7 * 7 * 20 cm^2

= 880 cm^2

Area of the base and the top    = (2`pi`r2)

= 2 * (22 / 7) * 7 * 7 cm^2

= 308 cm^2

Total surface area = (2`pi`r h) + (2`pi`r2) = 880 cm^2 + 308 cm^2

= 1188 cm^2

Example 2:

Surface area of a right circular cylinder of height 35 cm is 121 cm^2. find the radius of the base.

Solution:

Curved surface area = (2`pi`r h) = 121 cm^2

2*(22/7)* r *35 = 121

r = (7*121)/(2*22*35)

= 11/20

= 0.55 cm

Hence, radius of the base = 0.55 cm.

Example 3:

Curved surface area of a right circular cylinder is 6.6 m^2. if the radius of the base of the cylinder is 1.4 m , find the height.

Solution:

Curved surface area = (2`pi`r h) = 6.6

= 2 * (22/7) * 1.4 * h

= h = (7*6.6) / (2*22*1.4)

= (7*66) / (2*22*14)

= 3/4 = 0.75

Required height of the cylinder = 0.75 m.

Please express your views of this topic Surface Area of Cylinder by commenting on blog.

Geometry Practice Exam Problems:


1. the circumference of the base of a right circular cylinder is 220cm. if the height of the cylinder is 2m, find the lateral surface area of the cylinder.

2. A closed circular cylinder has diameter 20cm and height 30 cm. find the total surface area of the cylinder.

3. the radius of the base of a closed right circular cylider is 35 cm and its height is 0.5m. find the total surface area of the cylinder.

Answers:

1. 4.4 m^2

2. 2512 cm^2

3. 18700 cm^2

Construction of Triangles

Introduction to construction of triangles:

The triangles can be constructed if the following requirements are given such as follows,

The measurement of three sides should be given (or)

The measurement of the two sides and the included angle should be given (or)

The measurement of a side and any two angles should be given.

Now we are going to see about the construction of triangles. Is this topic Scalene Triangles hard for you? Watch out for my coming posts.


Construction of triangles:


Construction of triangles if three sides are given:

Construct a triangle if three sides are given with x, y and z measurements.

Steps of construction:

First a line segment QR of x cm length should be drawn.

With Q as center and radius of y cm be drawn and it equals to PQ and draw an arc of a circle.

With R as center and radius of PR = z cm and draw an arc and it will intersects at the first arc of point P.

Now join the points of line segments PQ and PR.

Thus, PQR is a required triangle. I have recently faced lot of problem while learning Geometry Definition, But thank to online resources of math which helped me to learn myself easily on net.

Other constructions of triangles:

Construction of triangles if two sides and angle are given:

Construct a triangle if two sides and an angle are given.

Steps of Construction:

First we have to draw a ray of QX of some length.

With the help of protractor measure the given angle and draw the line to meet Q.

The ray QY which may cut line segment QR of x cm.

The ray QY which may cut the line segment QP of y cm.

Now we can join the two points P and R.

Thus, PQR is the required triangle.

Construction of triangles if two angles and Side are given:

Construct a triangle if two angles and a side are given.


Steps of Construction:

First we should draw the line segment of QR of given length.

With the help of the protractor measure the given angle at RQX

Then, draw QRY for the given angle such that XY lie on the same side of the PQ.

Then, label the point where it intersects at QX and QY as P.

Thus, the PQR is the required triangle.

Base Pentagon Prism

Introduction:-
In geometry, the pentagonal prism is a prism with a pentagonal base.

Pentagonal prism is a type of heptahedron.

If faces are all regular, the pentagonal prism is a semi regular polyhedron and the third in an infinite set of prisms formed by square sides and two regular polygon caps.

A pentagonal prism has                                                                                                                                                                         Source:- Wikipedia.

7 faces
10 vertices
15 edges.
The pentagonal prism looks like .


Formulas For Pentagonal Prism:-


`Base area = 5/ 2 * a * s`
`Base perimeter = 5s`
`Prisms Surface area = 5as + 5sh.`
`Volume of Prism = (5/2)ash`
`here`
`a = apothem Leng th,`
`s = side,`

`h = height.`

Please express your views of this topic Congruence of Triangles by commenting on blog.

Problems on Prism:-


Problem 1:-

Find the base area  and base perimeter of the pentagonal prism if the apothem length is 4 and the side is 2.

Solution:-

Given

The apothem, length is 4

The side is 2.

The formula that is used to calculate the area of the base is `5/ 2 a* s.`

By plugging in the values of  a and s in the formula we get

Area of the base = `5/ 2` * 4 * 2.

By crossing 2 and 2 in the above equation we get

Area of the base = 5 * 4 = 20.square units.

The formula that is used to find the base perimeter is  `5s` .

By plugging in the given values we get

Base perimeter = 5 * s = 5 * 2 = 10.

Problem 2:-

Find the base area  and base perimeter of the pentagonal prism if the apothem length is 6 and the side is 4.

Solution:-

Given

The apothem, length is 6

The side is 4.

The formula that is used to calculate the area of the base is` 5/ 2 a* s.`

By plugging in the values of  a and s in the formula we get

Area of the base = `5/ 2 * 6 * 4.`

By crossing 2 and 4 in the above equation we get

Area of the base = 5 * 12 = 60.square units.

The formula that is used to find the base perimeter is  `5s.`

By plugging in the given values we get

Base perimeter = 5 * s = 5 * 4 = 20.

Problem 3:-

Find the base area and base perimeter of the pentagonal prism if the apothem length is 5 and the side is .

Solution:-

Given

The apothem, length is 5

The side is 7.

The formula that is used to calculate the area of the base is `5/ 2 a* s.`

By plugging in the values of  a and s in the formula we get

Area of the base = `5/ 2 * 5 * 7.`

Area of the base = `175/ 2` square units.

The formula that is used to find the base perimeter is  5s.

By plugging in the given values we get

Base perimeter = 5 * s = 5 * 7 = 35.

angles at a point

Introduction (Angles at point):

In geometry an angle is the figure produced by two ray’s distribution a common endpoint, called the vertex of angle. The degree of the angle is the quantity of revolution that separates the two waves, and deliberate by considering the length of circular curve is out when one ray is rotate regarding the vertex to correspond with the other. The angle along with a line and a curve or along with two intersecting curve.


Positive and negative angles at a point:

In mathematical script is that angles specified a sign are positive angles if considered anticlockwise and negative angles ? is efficiently the same to a positive angle of one full rotation less ?. if considered clockwise, from a known line. If no line is specified, that can be understood to be the x-axis in the Cartesian plane. In many geometrical situations a pessimistic angle of ?? is efficiently the same to a positive angle of one full rotation less ?.

Example, a clockwise rotation of 45° (angle of ?45°) is efficiently the same to an anticlockwise rotation of 360° ? 45° (angle of 315°).

Types of Angles:

Right angle
Acute angle
obtuse angle
reflex angle
Vertical opposite angles
Co-responding angles and Alternative angles
Interior angle
Identifying angles:

Angles may be recognized by the labels involved to the three points to identify them. Example, the angle by vertex A with this by the rays AB and AC.

Potentially, an angle denoted,  ?BAC may refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle as of C to B, or the anticlockwise angle as of C to B, wherever the way in that the angle is deliberate determines its sign.


Examples for angles at a point:


Example 1:

Find the value of x.

Solution:

x + 80° + 2x + x = 180° (contiguous angles on a straight line)

4x = 180° - 80°
= 100°

x = 100°
4
The answer of x = 25°

Example 2:

Find the value of x.

Solution:

48° + 90° + 120° + x = 360° ( Angles at a point )
x = 360° - (48° + 90° + 120° )
= 360° - 258°
The answer of x= 102°

Solve Allied Angles Axiom

Introduction to solve allied angles axiom:

The allied angles are nothing but the co-interior angles where they are transversely cuts the two parallel lines and the allied angles will be formed. The supplementary angles present in the geometry figures are also called as one of the types of allied angles. The allied angles total measurements are about 180 degrees. The tutors will describe the concepts to students with some example problems. Now we see how to solve allied angles axiom with the help of the tutor.

About How to Solve Allied Angles Axiom:

Now we see about how to solve for the allied angles axiom and its concepts with the help of the tutor. The allied angles are nothing but the angles where they are formed when two parallel lines are cut by a transverse line and the interior angles forms are called as the allied angles. The allied angles measurements are be about 180 degrees.

From the above figure it is given that as there are two parallel lines such as L1 and L2. These two lines can be cut by the traversal line T and the allied angles forms in the figure are given as a, b, c, d.

`angle a` +`angle d` = 180 degrees.

`angle b`  +`angle c` = 180 degrees.

The angle measures are given and we can determine the other angle with the total measurement of the angles are about 180 degrees.

Now we see some of the problems on allied angles axiom with the help of the tutor. Understanding Alternate Interior Angle is always challenging for me but thanks to all math help websites to help me out.

Problems to Solve Allied Angles Axiom:
Example:

Calculate the allied angles from the given figure?

Solution:

Now we see how to solve the allied angles measurement as follows,

The allied angles measurements are about 180 degrees.

From the given diagram, S1and S2 are the two parallel lines and cut by the transverse line T.

The allied angle are thus formed between the parallel lines.

For calculating the allied angle we have to do as follows,

Y + 70 = 180

Y = 180 - 70

Y = 110

Thus, the allied angles for 70 is about 110 degrees.

11th Grade Geometry

Introduction of 11th grade geometry :-

In grade (11) means eleventh grade is a year of education in all over the world. The eleventh grade is the final year of the secondary school. Students are usually 16 - 17 years old. Geometry is concerned with size, shape, relative figures etc. In 11th grade geometry lessons study about the angle, circle , quadrilateral etc.

Example Problems for 11th Grade Geometry :-

Problem 1:-

Determine the equation of the straight line passing through the points (1, 2) and (3, − 4).

Solution:

The equation of a straight line passing through two points is

`(y - y1)/ (y1 - y2)` = `(x - x1)/( x1 - x2)`


Here (x1, y1) = (1, 2) and (x2, y2) = (3, − 4).

Substituting the above, the required line is

`(y - 2)/(2 + 4)` = `(x - 1)/(1 - 3)`

`(y - 2)/6` = `(x - 1)/(- 2)`

`(y-2)/3` = `(x-1)/(-1)`

y − 2 = − 3 (x − 1)
y − 2 = − 3x + 3

3x + y = 5 is the required equation of the straight line.Is this topic AAA Postulate hard for you? Watch out for my coming posts.

Problem 2:-

Find the equation of the straight line passing through the point (1,2) and making intercepts on the co-ordinate axes which are in the ratio 2 : 3.

Solution:-

The intercept form is

`x/a +y/b` = 1 … (1)

The intercepts are in the ratio 2 : 3  a = 2k, b = 3k.

(1) becomes

`x/(2k) +y/(3k)` = 1     i.e. 3x + 2y = 6k

Since (1, 2) lies on the above straight line, 3 + 4 = 6k i.e. 6k = 7

Hence the required equation of the straight line is 3x + 2y = 7


Problem 3:-

Find the distance between the parallel lines 2x + 3y − 6=0 and 2x + 3y + 7 = 0.

Solution:-

The distance between the parallel lines is

`|(c_1 - c_2)/sqrt(a^2 + b^2)|` .

Here `c_1` = − 6, `c_2` = 7, a = 2, b = 3

The required distance is

`|(- 6 -7)/sqrt(2^2 + 3^2)|` = `| (-13)/sqrt(13)|`

= `sqrt(13)` units.


Practice Problems for 11th Grade Geometry :-

Problem 1:-

Find the equation of the straight line, if the perpendicular from the origin makes an angle of 120° with x-axis and the length of theperpendicular from the origin is 6 units.

Answer: The required equation of the straight line is x − `sqrt(3)` y + 12 = 0


Problem 2:-

Find the points on y-axis whose perpendicular distance from the straight line 4x − 3y − 12 = 0 is 3.

Answer: The required points are (0, 1) and (0, − 9).

Sum of Two Squares

Introduction to sum of two squares

In algebra, we have some formulae to expand squares.

They are:

`( a + b ) ^2 = a^2 + 2ab + b^2`
`( a ** b ) ^2` = `a^2 ** 2ab + b^2`
`( a + b ) ^2 + ( a ** b ) ^2` = `2 ( a^2 + b^2 )`
`( a + (1/a) ) ^2` = `a ^2 + (1/a^2) + 2`
`( a ** 1/a ) ^2` = `a ^2 + 1/a ^2 ** 2`
`( a + (1/a) ) ^2` + `( a ** 1/a ) ^2`   = `2 ( a ^2 + (1/a ^2))`
`a + b = sqrt (( a ** b ) ^2 + 4 ab)`
`a ** b ` = `sqrt (( a + b ) ^2 ** 4 ab)`


Keeping these formulae in mind, we can break up the square values to get the final answer. Now let us see few problems on sum of two squares. I like to share this Example of Obtuse Angle with you all through my article.

Example Problems on Sum of Two Squares

Ex 1: Find the value of a ^2 + b^2, If  a + b = 7 and ab = 7.

Soln: By using the above formulae, `a^2 + b ^2 = (1/2) [ ( a + b ) ^2 + ( a ** b ^2]`

Therefore  a – b = `sqrt (( a + b ) ^2 **4 ab)`

a – b = `sqrt (7 ^2 ** 4 ( 7 ))` = `sqrt (49 **28)` = `sqrt 21`

Therefore `a ^2 + b ^2` = `(1/2)[ ( a+ b ) ^2 + ( a ** b) ^2]`

= `(1/2) [ 7^2 + ( sqrt21)^2]`   =  `(1/2) [ 49 + 21 ]`

Therefore  `a^2 + b^2 = 35`

Ex 2: Find the value of A^2 + b^2, if a – b = 7 and ab = 18.

Soln: By using the above formula, `a + b = sqrt (( a ** b) ^2 + 4 ab)`

= `sqrt ((7) ^2 + 4 (18)) = sqrt (49 + 72)`

= `sqrt 121`    = 11

Therefore `a^2 + b^2` = `(1/2) [( a + b ) ^2 + ( a ** b ) ^2]`

= `(1/2) [ 11^2 + 7 ^2 ] = (1/2)[ 121 + 49 ]`

= `(1/2)` [ 170 ]  =  85

Therefore `a^2 + b^2 = 85`

Ex 3: If `a + (1/a) = 6` , find the value of `a ^2 + (1/a^2)` .

Soln: `a ^2 + (1/a^2) = (a + (1/a) ) ^2 **2 = ( 6 ^2) ** 2 = 34`

Therefore `a^2 + (1/a^2) = 34` [By using formula in 4]

Ex 4: If `a ** (1/a) = 8` , find the value of `a^2 + (1/a ^2)`

Soln: Therefore `a^2 + 1/a^2` = `(a ** (1/a)) ^2` + 2

= `8^2 + 2`

= 64 + 2 = 66.

Therefore `a^2 + (1/a^2)` = 66.   [By using formula in 5]

Ex 5: If a^2 – 5a – 1 = 0, find the value of `a^2 + (1/a^2)`

Soln: Given:  a2 – 5a – 1 = 0

`rArr` a – 5 – (1/ a) = 0    [Divide throughout by]

`rArr` `a **(1/a)`  = 5

Therefore `a^2 + (1/a^2)`  = `(a ** (1/a) ) ^2` + 2 =  `5^2` + 2 = 27. Understanding Area of Hexagon is always challenging for me but thanks to all math help websites to help me out.

Practice Problems on Sum of Two Squares

1. If a + 1/a  = 2, find a^2 + 1/a^2

Ans: 2

2. If a + b = 9 and ab = -22, find the values of a ^2 + b^2.

[And: 125]

3. If a^2 – 3a + 1 = 0, find the value of a^2 + 1/a ^2.

[Ans: 7]

Set-builder Notation Online Help

Introduction:

Set-builder notation is a mathematical data for concerning a set by stating the property that the members of the set must suit. The terms "things" in a set is known as the "elements", and are listed inside curly braces. In online study, many websites provides free help on set builder notation questions. In online study, students can learn more about all topics. Practicing set-builder notation problems help students to get prepared for test preparation and exam preparation. Practicing these problems helps student to get good scores in test and exams. In this article, we are going to see about, set-builder notation online help.I like to share this Surface Area Triangular Prism with you all through my article.

Set-builder Notation Online Help: - Representation of Set-builder Notation

The set {x: x < 3} is stated as, "the set of all x such that x is less than 0." The other form of representation of set-builder notation is the vertical line, {x | x < 3}.

General Form: {formula for elements: restrictions} or {Method for elements| restrictions}

{X: x ≠ 5} the set of all real numbers except 5

{X | x < 15} the set of all real numbers less than 15

{X | x is a positive integer} the set of all real numbers which are positive integer

{n + 1: n is an integer} The set of all real numbers (e.g. ..., -2, -1, 1, 2, 3...)

Understanding help on math homework for free is always challenging for me but thanks to all math help websites to help me out.

Set-builder Notation Online Help: - Examples

Example 1:

Express the following in set-builder notation: Y = {45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60}

Solution:

The set-builder notation for the given problem is given as,

Y = {45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60}

{ Y| Y `in` R, 44 < Y < 61}

Example 2:

Write the set of even numbers between 10 and 30 in set-builder notation.

Solution:

The set-builder notation for the given problem is given as,

X = {10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}

{X|X `in`   even numbers, 10 `<=` X `<=` 30 }

Vertex Cycle Cover

Introduction to vertex cycle cover:

Vertex cycle cover is defined as the number of cycles, which are having the vertices and edges. The vertices are represented as G. Vertex graph are also having the subgraph.This subgraph also represented using the letter G.In this vertex cycle cover, each cover of the cycle are having only one cycle. The length of the cycle are also mentioned in this vertex cover cycle.

Explanation for Vertex Cycle Cover:

Vertex cycle cover are having the subgraph and the vertices. In this vertex cycle cover, if no vertices are present in common means, then that cycle is called vertex-disjoint cycle.If the cycles are having no edges present means, then that cycle is called as the edge-disjoint cycle. Vertex cycles covers having short cycles covers are used to represent the cubic graph. This can also having the applications in the permanent and the minimum cycles.

Properties of Vertex Cycle Cover:

The properties of the vertex cycle graph are mentioned below the following,

1. Vertex cycle cover is a permanent one among the remaining vertex covers.

The permanent vertex cycle cover are having both the directed graph and also the adjacent matrix. Both of them are mentioned in the vertex cycle cover.

2.Vertex cycles covers are having only minimum disjoint cover cycles.

This vertex cycle graph are having only minimum disjoint cycles.Because these are mentioned in the problem of finding the complexity of vertex.

3. Vertex cycles covers are having only minimum weight cover cycles.

This vertex cycle cover having the minimum weight covers are denoted by using the weighted graph present in the vertex gaph.This minimum weight cover cycles are having the sum of weights for the respective vertices.

4.Vertex cycles covers are having only double cover cycles.

This vertex cycle cover problem having the double cycle cover are denoted by using the open cycles. The set of vertices representing the open cycles.

Measure Square Yards

Introduction to Square Yard:-
The square yard is an imperial/US customary (non-metric) unit of area, formerly used in most of the English-speaking world but now generally replaced by the square metre outside of the U.S., Canada and the U.K. It is defined as the area of a square with sides of one yard in length. I like to share this Surface Area of a Trapezoidal Prism with you all through my article.

Measure Square Yards-solved Problems:-

area = `a^2` .

a = length of side.

Here a = 4.

By plugging it in to the formula we get

`area = 4^2 `  = 16.

So the area of the square is 16 square yards.

Problem 2

Measure the area of the rectangle which has the length of 5 yard and breadth 3 yard.

Solution:-

Given the length and breadth of the rectangle is 5 yard and 3 yard respectively.

The formula used to find the area of rectangle is   length * breadth.

Area= Length*breadth

By plugging in the given values in to the formula we get

Area = 15 * 9

= 145

Area of the given rectangle is 145 square yards.

Problem 3

Measure the area of the circle which has the radius of 6 yards.

Solution:-

Given the radius of the circle is 6 yards.

The formula used to find the area of the circle is `pi r^2` .

Area = `pi r^2` .

By plugging in the given values in to the formula we get

Area = `pi 6^2`

= 36 `pi` .

Area of the given circle is 36 pi squareyards. Please express your views of this topic completing the square equation by commenting on blog

Measure Square Yards-practice Problems:-

Problem: - 1

Measure the area of the rectangle which has the length of 10 yards and breadth 8 yards.

Answer: - 80 square yards.

Problem: - 2

Measure the area of the square with side measure of 4 yard find the it by applying the conversion of yard to yards.

Answer: - 144 square yards.

Problem: - 3

Measure the area of circle which has the radius of 2 yards.

Answer: - square yards.

Naming Lines in Geometry

Introduction for naming lines in geometry:
Lines are one dimension straight geometry figure and in solid geometry lines are used in designs.A lines are start with the one end and end with one direction then it said to be line segment.Lines are classified into many types which depends upon the line projection.Line segment is denoted with a connected piece of line.line segments names  has two endpoints and it is named by its endpoints. In this article contains naming lines in geometry I like to share this Area of a Rhombus Formula with you all through my article.

Naming Lines in Geometry:

In naming lines geometry section we have many types of lines which has propertyof its own.Lines are classified into following types.

Parallel lines:
In geometry parallel lines are mostly aplicable in design section, two lines which does not touch each other are called parallel lines.

Perpendicular lines:
In geometry Perpendicular lines are mostly aplicable drawing section,Two line segment  that form a L shape are called perpendicular lines.

Intersecting lines:
If two lines intersect at a point, these lines are called intersecting lines.

Concurrent lines:
The three or more lines passing through the same point are called concurrent lines. Understanding math help live is always challenging for me but thanks to all math help websites to help me out.

Problems in Naming Lines Geometry:

Example 1:
Find co-ordinate of the mid point of the line segment joining given points A(-4,1) and B(5,4)

Solution:
The required mid point is
Formul a   `((x_1+x_2)/2 ,(y_1+y_2)/2)` here,  (x1, y1) = (-4,1),(x2, y2) = (5,4)

=  `((-4+5)/(2))``((1 +4)/(2)) `

= `(-1/2) ` ,  ` (5/2)`


Example 2:
Find the slope of the lines given (1,-3) and (-1,3)

Solution:
(x1,y1)= (1,-3), (x2,y2)= (-1,3).
We know to find slope of line,m=` (y_2-y_1) /(x_2-x_1)`

=`(3+3)/(-1-1)`

m =`6/-2` = -3

Example 3:
Find the equation of the line having slope 5 and y-intercept -1.

Solution:
Applying the equation of the line is y = mx + c
Given,       m =5 ,c = -1
y = 5 x -1

or  y = 5x - 1
or  5x- y +1 = 0.

Plot Points

Introduction:

A rectangular co-ordinate system, or Cartesian plane, is a set of two intersect and vertical axes forming a xy plane. The horizontal axis is generally labelled the x-axis and the vertical axis is generally labelled the y-axis. The two axes split the planes into four parts known as quadrants. Any point on the plane communicate to an ordered pair (x, y) of valid numbers x and y.

Types of Plot Points:

Line plot
Scatter plot
Stem and Leaf Plot
Box plot
Line plot: A line graph plots constant data as points and then joins them with a line. Multiple data sets can be graphed simultaneously, but a key have to be used.

Scatter plot: A scatter plot defined as the organization between the two factors of the testing. A line which is used to find the positive, negative, or no correlation.

Stem and Leaf Plot: Stem and leaf plot points are defined as the documentation data values in rows, and can easily be made into a histogram. Large information sets can be accommodated by splitting stems.

Box plot: A box plot points are defined as a concise graph screening the five point abstract. Multiple box plots can be drawn side by side to evaluate more than one information set.

Advantages and Disadvantages of Plot Points:

Line plot

Advantages:

Immediate analysis of data.

Shows variety, minimum & maximum, gaps & clusters, and outliers simply.

Accurate values retained.

Disadvantages:

Not as visually attractive.

Top for below 50 data values.

Desires small range of data.

Scatter plot

Advantages:

Shows a movement in the data connection.

Retains accurate data ideals and example size.

Shows lowest/highest and outliers.

Disadvantages:

Hard to imagine outcome in huge data sets.

Flat drift line gives indecisive results.

Stem and Leaf Plot


Advantages:

Concise symbol of data.

Shows range, smallest & highest, gaps & clusters, and outliers simple.

Can hold very large data set.

Disadvantages:

Not visually attractive.

Does not simply indicate events of centrality for huge data sets.

Box plot

Advantages:

Shows 5-point review and outliers.

Simply compare two or supplementary information sets.

Handles extremely large data sets easily.

Disadvantages:

Not as visually attractive as extra graphs.

Accurate values not retained.

Surface Area of a Box

Introduction to surface area of a box:
Box is same as the cuboid. In box the dimensions are length, width and height. If all the dimensions are equal then the box is cube and if it is different then the box is cuboid. Box is a 3 dimensional image. A box has 8 vertices, 12 edges and 6 faces. Understanding Area of a Hexagon is always challenging for me but thanks to all math help websites to help me out.

Diagram and Formula – Surface Area of a Box:

Surface area of a box = 2(1w+lh+wh)

Where w`=>` width of the box

h`=>` height of the box

l`=>` length of the box. Is this topic how to construct parallel lines hard for you? Watch out for my coming posts.

Example Problems – Surface Area of a Box:

Example 1 :

Find the surface area of a box whose length, width and height are 12cm, 14cm, 16cm.

Solution:

Given that,

Length of the box = 12cm

Width of the box = 14cm

Height of the box=16cm.

Surface area of a box = 2(lw + lh + wh)

=2((12*14) +(12*16)+(14*16))

=2(168+192+224)

=2(584)

=1168cm3.

Example 2 :

Find the surface area of a box whose length, width and height are 2cm, 4cm, 6cm.

Solution:

Given that,

Length of the box = 2cm

Width of the box = 4cm

Height of the box=6cm.

Surface area of a box = 2(lw + lh + wh)

=2((2*4) +(2*6)+( 4*6))

=2(8+12+24)

=2(44)

=88cm3.

Example 3 :

Find the surface area of a box whose length, width and height are 8cm, 6cm, 4cm.

Solution:

Given that,

Length of the box = 8cm

Width of the box = 6cm

Height of the box=4cm.

Surface area of a box = 2(lw + lh + wh)

=2((8*6) +(8*4)+(6*4))

=2(48+32+24)

=2(104)

=208cm3.

Example 4 :

Find the surface area of a box whose length, width and height are 10cm, 20cm, 30cm.

Solution:

Given that,

Length of the box = 10cm

Width of the box = 20cm

Height of the box=30cm.

Surface area of a box = 2(lw + lh + wh)

=2((10*20) +(10*30)+(20*30))

=2(200+300+600)

=2(1100)

=2200cm3.

Two Parallel Lines Cut by a Transversal

Introduction to two parallel lines cut by a transversal:

Parallel lines:

Two lines on a plane that never  intersect or meet is known as parallel lines. The distance between both the lines must be the same. And they must not intercept with each other. It can also be explained as the length between parallel lines will be exactly same at any point. Examples of parallel lines are railway track, etc.

Transversal:

A straight line is said to be transversal if the line cuts two or more parallel lines at different points. In the figure the line l cuts the parallel lines a and b. So the line l is called as a transversal line

Two Parallel Lines Cut by a Transversal:

When two parallel lines are cut by a transversal then

The corresponding angles are equal

Pair of Vertically Opposite angles is equal

Pairs of Alternate interior angles are equal

Interior angles on same side are supplementary

Conditions Satisfies when Two Parallel Lines Cut by a Transversal:

The corresponding angles formed by the transversal will be equal. For example angle 4 and angle 6 are corresponding angles, and the other pairs of corresponding angles are (5 and 3), (8 and 2) and (1 and 7).

Therefore we have:               angle 6 = angle 4

angle 5 = angle 3

angle 8 = angle 2

angle 7 = angle 1

Pair of Vertically Opposite angles is equal

The pair of vertically opposite angles is equal when a transversal line is formed. For example, angle 1 and angle 3 are vertically opposite angles and the other pairs of vertically opposite angles are (2 and 4), (5 and 7) and (6 and 8).

Therefore we have:                angle 1 = angle 3

angle 2 = angle 4

angle 5 = angle 7

angle 6 = angle 8

Pairs of Alternate interior angles are equal

The pair of alternate angles is equal when the transversal line is formed. Here the pairs (2 and 6) and (3 and 7) are alternate interior angles. I have recently faced lot of problem while learning simple math problems for kids, But thank to online resources of math which helped me to learn myself easily on net.

Therefore we have:                angle 2 = angle 6

angle 3 = angle 7

Interior angles on same side are supplementary

Interior angles on same side are supplementary when a transversal is formed. The angles on the same side of the transversal are (6 and 3) and (2 and 7).

Therefore we have:                angle 3+ angle 6 = 180

angle 2 + angle 7 = 180

Radius from Circumference

Introduction to radius from circumference:
In day to day life, we often came across some unique math terms. Radius is one of the special math terms that falls under this category.

Radius of a circle is nothing but the line segment from the center of the circle to its perimeter. In other terms, half the diameter is the radius.

In this article of radius from circumference, we are going to find the radius of the circle from the circumference formula.

Formula for Radius from Circumference:

The Circumference of the circle is given by the following formula:

Circumference =  2`pi`r

If the Circumference C is given, the radius can be calculated by the following formula:

radius  =  `C/(2 pi)`

Example Problems for Finding Radius:
Example 1:

Find the radius, if the circumference of the circle is 100 cm

Solution:

Radius of circle, r  =  `C / ( 2 pi )`

=  `100 / (2 * 3.14)`

=  `100 / 6.28`

=  15.92 cm

Example 2:

Find the radius, if the circumference of the circle is 94 cm

Solution:

Radius of circle, r  =  `C / ( 2 pi )`

=  `94 / (2 * 3.14)`

=  `94 / 6.28`

=  14.97 cm

Example 3:

Find the radius, if the circumference of the circle is 60 mm

Solution:

Radius of circle, r  =  `C / ( 2 pi )`

=  `60 / (2 * 3.14)`

=  `60 / 6.28`

=  9.55 mm

Example 4:

Find the radius, if the circumference of the circle is 35 cm

Solution:

Radius of circle, r  =  `C / ( 2 pi )`

=  `35 / (2 * 3.14)`

=  `35 / 6.28`

=  5.57 cm

Example 5:

Find the radius, if the circumference of the circle is 20 mm

Solution:

Radius of circle, r  =  `C / ( 2 pi )`

=  `20 / (2 * 3.14)`

=  `20 / 6.28`

=  3.18 mm

Example 6:

Find the radius, if the circumference of the circle is 4 m

Solution:

Radius of circle, r  =  `C / ( 2 pi )`

=  `4 / (2 * 3.14)`

=  `4 / 6.28`

=  0.64 m


Practice Problems for Finding Radius:

1) Find the radius, if the circumference of the circle is 50 cm

Answer: 7.96 cm

2) Find the radius, if the circumference of the circle is 20 m

Answer: 3.18 m

3) Find the radius, if the circumference of the circle is 110 mm

Answer: 17.52 mm

4) Find the radius, if the circumference of the circle is 75 cm

Answer: 11.94 cm

5) Find the radius, if the circumference of the circle is 72 mm

Answer: 11.46 mm

Geometry Edge of Rectangular

Introduction to Geometry edge of rectangular:

Rectangular shape is one of the geometry two dimensional objects. Geom`etry edge of rectangular properties are the crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. Its angles are not right angles. Opposite sides are parallel and congruent . The diagonal bisect each other The diagonals are congruent. A four -sided plane figure with four right angles. Understanding Volume of a Rectangular Prism Formula is always challenging for me but thanks to all math help websites to help me out.

Basic Concepts of Geometry Edge of Rectangular:

Geometry edge of rectangular:

Each and every object should have edges. Two edges make the angle of geometry object.And also edges to make the corners and vertices of object .Rectangle have  four  edges and also have four vertices or corners .Each corner make the angle of 90degree.

From this diagram:

AB, BD, DC, AC are edges of the rectangle.

AB edge is parallel to CD edge

AC edge is parallel to BD edge

AB || CD, AC || BD (opposite sides are equal in rectangle)

Each edge should make the angles are


Please express your views of this topic how many edges does a rectangular prism have by commenting on blog.

Area and Perimeter of the Geometry Edge of Rectangular:

Area and perimeter:

Area of rectangle= Length * Width

Perimeter of the rectangle=2(Length + Width)

Both area and perimeter are depends on the rectangle edges.

A rectangle has two Length edges and two width edges.

Length edges are AB, CD

Width edges are BD, AC

Two length edges are equal AB=CD

Two width edges are equal BD=AC

Example problems in Geometry edge of rectangular:

1.From given diagram Find the area and perimeter of the rectangle  and name of the rectangular edges?

Solution:

Given data PQRS is closed rectangle

PQ= 13cm(PQ||SRsO P)

PQ=SR=13

QR=5.5cm(QR||PS)

So QR=PS=5.5

Length   of the rectangle=13

Width of the rectangle =5.5

Finding the area:

(1  )Area of the rectangle=Length x Width

= 13x5.5

Area =71.5cm2

(2) Finding the perimeter:

Perimeter of the rectangle=2( L+W)

=2(13+5.5)

=2(18.5)

=37cm

Perimeter=37

(3)Finding the name of the edges:

From given diagram edges are PQ,QR,RS,SP

Volume of Equilateral Triangle

Introduction

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons, and can therefore also be referred to as regular triangles.As, equilateral triangle is two dimensional, it is impossible to determine its volume.So, lets know how to determine volume of the prism with equilateral triangular base.

(Source -wiki)

Formula for Volume of Equilateral Triangle Base Prism:

The volume of equilateral triangle, ` v = 1/3 B xx H`

Example for Volume of Equilateral Triangle Base :

Example 1. Find the volume of equilateral triangle base prism of side 10 cm and height `17/2 cm`

Solution:

Let, H = `17/2 cm` , B = 10cm

Formula used:  ` V = 1/3 B H`

Therefore, volume of equilateral triangle base prism ,

`V = 1/3 xx 10 xx 17/2` = `85/3 cm3`

Example2. Find the volume of equilateral triangle base prism of side 100 m and height 85 m

Solution:

Let, `H = 17/2 m` , B = 10 m

Formula used: ` V = 1/3 B H`

Therefore, volume of equilateral triangle base prism,

`V = 1/3 xx 10 xx 17/2`

= 2833.33 m3

Example3. Find the volume of equilateral triangle base prism of side 9 cm and height `15/2` cm

Solution:

Let, `H = 15/2 cm` , B = 9 cm

Formula used:   `V = 1/3 B H`

Therefore, volume of equilateral triangle base prism ,

`V = 1/3 xx 9 xx 15/2`

= `45/2` cm3

Example4. Find the volume of equilateral triangle base prism of side 90 cm and height 75 cm

Solution:

Let, H = 75cm, B = 90 cm

Formula used:      `V = 1/3 B H`

Therefore, volume of equilateral triangle base prism ,

`V = 1/3 xx 90 xx 75`

= 2250 cm3



Practice Problems Volume of Equilateral Triangle Base Prism

1. Find the volume of equilateral triangle base prism of side 180 m and height 156 m

2. Find the volume of equilateral triangle base prism of side 450 cm and height 390 cm

Solution for practice problems of volume of equilateral triangle base prism :

1. 9360 m3

2. 58500cm3

Understanding the Concepts of Set Theory Using Venn Diagram

Sets can be termed as a collection of objects. The objects must be similar in nature. In mathematics, the objects must be related to the mathematical world. The set theory can be understood with the help of diagrammatically explanation. The Venn diagram example can be used for this purpose. The Venn diagram questions relate to the set theory and can help one to understand the set theory better. So, basically set theory can be best understood with the help of these. The definition of Venn diagram states that they are diagrams depicting sets and the operations on them. The operation on sets can be easily explained with the help of these. Once the definition has been understood the same concept can be used for the solving of the problems. The problems can be easily solved once the concept is clear.

The picture of a Venn diagram represents sets pictorially and helps in solving the operations on it. The various operations on sets include Union and intersection. The Venn diagram symbols have to be learnt first and then the whole thing can be implemented. There can problems on how to find the union or intersection of two sets. The union of sets will result in a set which will contain all the elements present in both the sets and the intersection will contain only those elements which are common to both the sets. The diagram will show the common area between the sets. This common area will denote the intersection of the sets. There can be also special type of sets in set theory. Some of them are empty set or unit set. When the intersection of two sets does not have a common area it means that there are no common elements between and the resultant set is an empty set. In the unit set there will be only one element.Please express your views of this topic how many faces does a triangular prism have by commenting on blog.

There can sets to include the various elements and can be named by the elements present in them. There are natural number sets or whole number sets. There are also sets containing the real numbers and the rational numbers. Complex numbers can also have sets in their name which will contain the collection of complex numbers. The square root of a negative number can be an example of a complex number. The number of elements in a set is denoted by the term called cardinality.