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)

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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