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.

I have recently faced lot of problem while learning math word problems for 7th grade, But thank to online resources of math which helped me to learn myself easily on net.


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.