Table of Contents

Lesson 3: Geometry. 2

3.1 Two-Dimensional Figure. 3

3.2 Three-Dimensional Figure. 6

3.3 What are Parallel Lines?. 9

3.4 What are Intersecting Lines?. 10

3.5 Congruent Figures. 11

3.6 Symmetry in Figures. 12

3.7 Line of Symmetry. 14

3.8 Coordinate Plane. 15

3.8.1 Parts of a Coordinate Plane. 16

3.9 Polygon. 19

3.10 Triangle. 20

3.11 Quadrilateral 21

3.11.1 Family Tree for Quadrilaterals. 22

3.12 Transformation. 23

3.12.1 Congruency of Figures by Transformation. 24

3.13 Sketching Transformations with Coordinate Plane. 25

3.14 Calculating Perimeter Using Grids. 26

3.15 Sum of Angles. 27

 


 

 

Lesson 3: Geometry

 

By the end of this lesson, you will be able to:

         Know about Coordinate plane

         Sketch the transformation of figures

         Solve the problems related to sum of angles.

 


 

3.1 Two-Dimensional Figure

 

Figures that can lie on a plane surface are known as Two-Dimensional figures.

They have only 2 dimensions, length and width.

Figure

Image

Attributes

Perimeter

Triangle

 

h

 

 

b

3 sides

3 vertices

3 angles

Sum of all the three sides.

Square

a

 

a a

 

 

a

4 sides

4 vertices

4 angles

 

(4 a)

Rectangle

l

 


b b

 

 

l

4 sides

4 vertices

4 angles

 

2(l + b)

 

Circle

Oval:      r

0

 

(2 π r)

Trapezoid

b

 


a

 

 

c

4 sides

4 vertices

4 angles

 

Sum of all the sides.

Hexagon

t

6 sides

6 vertices

6 angles

(6 t)

 

 

Find the perimeter of the following figures:

Text Box: Example 1

1) 2)

7 cm 9 cm

 

4 cm 4

7 cm 7 cm

9 cm

1)    cm

1) The first figure is a square having all the sides equal.

         Side = 7 cm = a

         Perimeter = 4 a

         Thus, Perimeter = 4 7 = 28 cm

2) The second figure is a rectangle having length and breadth

         Length = 9 cm = l

         Breadth = 4 cm = b

         Perimeter = 2 ( l + b )

         Thus, Perimeter = 2 ( 9 + 4) = 2 13 = 26 cm

 

 

 

 

 

 

 

 

 

 

 

Text Box: Example 2
 


Find the perimeter and area of the following figures:

 

1)    2)
20 ft 25 ft 30 ft

Text Box: Kitchen   Garage            15 ft
                                     Hall
  Living Room                  35 ft
Trapezoid:   6 cm A 4 cm B

10cm 10cm 50 ft

 


D 18 cm C 45 ft 30 ft

1) The first figure ABCD is a Trapezoid.

         AB = 4 cm ; BC = 10 cm ; CD = 18 cm ; DA = 10 cm.

         Perimeter = sum of all the sides = ( AB + BC + CD + DA )

         Perimeter = ( 4 + 10 + 18 + 10 ) = 42 cm.

 

2) Consider the second figure which shows length of different sections of a house.

         Perimeter = sum of all outer boundaries = ( 20 + 25 + 30 + 15 + 35 + 30 + 45 +50 )

         Perimeter = ( 45 + 45 + 65 + 95 ) = 250 ft.

 


 

3.2 Three-Dimensional Figure

 

These are like solids. They cannot lie on a plane surface. They have length, width and height. It can be measured in 3 directions. They can be identified by faces, edges and vertexes.

 

3-D gives us depth perception.

 

         A face is a flat surface.

         An edge is a line formed by 2 faces.

         A vertex is a point where 2 or more edges meet.

 

The 3D figures which have curved surfaces as below, can be identified by curved surfaces and the flat circular surfaces (bases) on which they rest.

 

 

 

 

 

Figure

Image

Description

Square

Pyramid

 

 

h s

 


l

l

         5 faces

         8 edges

         5 vertices

 

Cube

 

a

 

 


a

a

         6 squre faces

         12 edges

         8 vertices

 

Cuboid

 

h

 

w

l

 

         6 faces

         12 edges

         8 vertices

 

Sphere

 

 

 

 

 


d

         1 curved surface.

         Radius = r =

         Diameter = d = 2 r

 

Cylinder

 

H h

 

r

         1 curved surface.

         2 circular bases.

         Radius = r; Heght = h

 

Cone

h l

 

r

         1 curved surface

         1 circular base

         Height = h; slant height = l; radius = r

 

 

 

 

 

 

 

 

Text Box: Example 3 Consider a cuboid as shown below. Find faces, edges and vertices?

Cube:                                                  G

        H
A B

 


F

 

C

 


E D

Faces = 6 ( ABGF, GBCD, HCDE, ABHC, FAHE, FGDE )

Edges = 12 (AB, BC, CD, DE, EF, FA, FG, GD, GB, AH, HE, HC )

Vertex = 8 ( A, B , C , D, E, F, G, H )

Text Box: Example 4 


Identify the given figures and structures:

1) Dice 2) Football

 

 

Cube:           DusterCan:     
     Juice
                         
3) Juice Can 4) Duster

 

 

 

 


The above figure has the following shapes:

Dice

Cube

Football

Spherical

Juice Can

Cylindrical

Duster

Cuboid

3.3 What are Parallel Lines?

 

Lines that are same distance apart at all points are known as Parallel Lines..

Parallel lines will never cross or intersect each other.

Line AB and CD are parallel lines.

 

 


B

A

D

C

 

Text Box: Example 5 


Identify the parallel lines from the given figures:

 

1) A B 2) P Q

 

R S

C D

 

 

From the above figures we conclude:

1.    In first figure line AB is parallel to line CD and line AC is parallel to line BD.

2.    In second figure line PQ is parallel to line RS and line PR is parallel to line QS.

 

 

3.4 What are Intersecting Lines?

 

Lines that crosses each other and have one point in common are known as Intersecting lines.

Let the two lines AB and CD intersect each other at P.

 

C B

P

 

A D

 

Perpendicular lines are special type of intersecting line which form right angle where they intersect.

Let us consider two lines PQ and RS which intersect at O.

S

 

O 90

P Q

 

 

R

 

Angle O makes 90 and is the common intersecting point of line PQ and RS.


 

3.5 Congruent Figures

 

Congruent figures are of same size and shape. When we place one figure over another, they should match exactly.

 

Shape

Image

Same Shape

Same Size

Rectangles

Yes

Yes

Circles

Yes

Yes

Rectangles

Yes

Yes

Quadrilateral

Yes

Yes


 

3.6 Symmetry in Figures

 

A figure is said to have symmetry if it can be folded to make 2 halves that exactly match each other. The two halves need to be mirror images of each other. We find symmetry everywhere in nature. Fruits, Flowers, Animals, Rainbow, Sun, Moon are some examples.

 


If we fold ourselves along the line in the middle, the two halves will match exactly

Butterfly is Symmetrical.

Sweet puppy is Symmetrical too. In fact all animals you will find to be symmetrical.

A

Text Box: Example 6 Below are few Symmetrical shapes

 

Text Box: Example 7        


There are other figures which are not symmetrical.

L

e

 

3.7 Line of Symmetry

 

The line along which we fold an object to get two equal parts is called Line of Symmetry. Any figure may have one or more Lines of Symmetry.

A figure having at least one Line of Symmetry is Symmetrical.

Text Box: Example 8

Consider some of the figures as shown below:

Figures

No. of Lines of Symmetry

Symmetrical

0

No

1

Yes

>1

Yes

 


 

3.8 Coordinate Plane

 

Suppose, you are standing in a room. Now if someone asks you, what is your location in the room? What would you say? You would try and estimate your distance from the door, walls etc and say that, I am 3 feet from the door or I am 3 feet from the door and 2 feet from the back wall. This is the origin of the coordinate system. We define our location at a place by an ordered pair of numbers.

Coordinate plane is a flat region determined by two perpendicular intersecting number line called axes. It is sort of a grid which is used to locate points. Each point is denoted by an ordered pair of numbers. We call each of these numbers the coordinates. Hence the name, coordinate plane. Here the floor of the room acts a coordinate plane.

 


 

3.8.1 Parts of a Coordinate Plane

 

         Origin: The starting point so to say. Coordinates are (0,0)

         X Axis: The horizontal number line marked x. Tells us how far right of the origin a point is.

         Y Axis: The vertical number line marked y. Tells us how far up from the origin a point is.

 

 


4

 

 

 

 

 

Text Box: Move     up

3

 

(2, 3)

 

 

 

 

2

 

 

 

 

 

 

 

 

 

0

 

3

 

 

 

 

x

 

Text Box: Move right

y

 

5

 

4

 

2

 

1

 

5

 

1

 

Origin (0, 0)

 

 

Here we have located the point (2,3) on the coordinate plane. It is 2 units from x axis and 3 units from y axis. So in the ordered pair, first number is the distance from x axis and the second number is the distance from y axis. They are named x coordinate and the y coordinate respectively.


 

Text Box: Example 9` Find the coordinates of Points A, B, C, D on the coordinate plane.

 

 

 


4

 

Text Box: A

 

 

 

Text Box: BText Box: Move     up

3

 

 

 

 

 

2

 

 

Text Box: C

 

 

 

 

 

 

0

 

3

 

Text Box: D

 

x

 

Text Box: Move right

y

 

5

 

4

 

2

 

1

 

5

 

1

 

Origin (0, 0)

 

 

 

Point

How far right to the origin (x coordinate)

How far up from the origin (y coordinate)

Coordinates

A

1

4

(1, 4)

B

0

3

(0, 3)

C

3

2

(3, 2)

D

2

0

(2, 0)

 

 


Point D Lies on x axis. Its y coordinate is 0.

Point B Lies on y axis. Its x coordinate is 0.

 

 

 

 

 

 

 

 

 

Text Box: Example 10 Find the coordinates of Pillow (P), Door (D), Boy (B) on the floor.

 

With reference to the given origin (0,0), moving on the x and y axis (direction shown), find the coordinates of the Pillow P, Boy B, Door D

Point

How far left to the origin (x coordinate)

How far forward from the origin (y coordinate)

Coordinates

P

5

0

(5, 0)

B

2

2

(2, 2)

D

0

1

(0, 1)

 

 


Notice that x is not to the right of origin but to the left here

Notice that y is below the origin but forward.

Here we just need to follow the direction of the arrow (x and y).


 

3.9 Polygon

 

Two- Dimensional closed figures with straight sides are called Polygons. Circle is a closed figure but is not a polygon.

Various types of Polygons:

Name

Image

Number of Sides

Number of Angles

Triangle

3

3

Quadrilateral

4

4

Pentagon

5

5

Hexagon

6

6

 

 


 

3.10 Triangle

 

Triangle is a polygon having 3 sides and 3 angles. Sum of the angles is 180.

 

Types of Triangles on the basis of sides

Equilateral Triangle

All 3 sides are equal

All three angles are 60

Isosceles Triangle

Two sides are equal

Two angles are equal

Scalene Triangle

No sides are equal

No angles are equal

 

 

Types of Triangles on the basis of angles

Right Triangle

One 90 angle

Acute Triangle

All angles < 90

Obtuse Triangle

One angle > 90

 


 

3.11 Quadrilateral

 

Any two dimensional shape with four straight sides is a quadrilateral

(Quad means four; lateral means side, hence quadrilateral)

         Four Sides

         Four Vertices

         Sum of interior angles equals 360 degrees.

 

Figure

Image

Attributes

 

 

Square

a

 


a a

 

 


a

4 equal sides.

Opposite sides are parallel

4 equal right angles (90)

 

Rectangle

l

 


b b

 

 


l

Opposite sides are equal and parallel

4 equal right angles (90)

Square is a rectangles with all sides equal

Parallelogram

Opposite sides are equal and parallel

Opposite angles are equal

 

Rhombus

4 equal sides

Opposite sides are parallel

Opposite angles are equal

Diagonals intersect at 90

Rhombus is a parallelogram with all sides equal.

Trapezoid/

Trapezium

b

 


a

 

 


c

One pair of opposite sides is equal

 

 

 

 

Kite

Two pair of adjacent equal sides

Angles where pairs meet are equal

Diagonals intersect at 90

 


3.11.1 Family Tree for Quadrilaterals

Text Box: Rectangle

 

 


 

3.12 Transformation

 

         Changing the position of a shape by rotation, reflection or translation is known as transformation.

         Size, area and length of the shape are unaltered.

         Turn, flip and slide are some of the basic moves.

         We can verify congruency of figure through these transformations.

 

Text Box: Example 11 


Consider an example as shown below:

 

 


Rotation B

 

 


A

 

( Turn)

C

 

A B

 

 

 

Turn

C

 

B

Reflection

 

 


A

 

( Flip )

C

 

 

 

B

 

 

 


A

 

 

C Flip

A C

Translation

 

( Slide)

 

 

 


B

 

 

 

 

 

 

 

 

A

C

 

Slide

 

B

 

3.12.1 Congruency of Figures by Transformation

 

If we only rotate, reflect or translate the shape then the two shapes are congruent

If we need to resize it then two shapes are similar

 

Rotate

Congruent

Reflect

Translate

 

 

         The size of the figure remains the same after turn, flip or slide.

         Two shapes of same size are said to be congruent when they can be super-imposed on one another.

         All congruent triangles are similar but all similar triangles are not congruent.

         Size can be different for similar triangles or species.

         Size should be exact same for congruency.

 


 

3.13 Sketching Transformations with Coordinate Plane

 

Reflection (Flip)

         Measure the distance of all the vertex points from the y axis.

         Mark the same distance on the other side of the y axis

         Join up the dots

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Translation (Slide)

         Move all the vertexes to equal distance, along the x axis (here 8 places)

         Join up the dots

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rotation (Turn)

We are doing here translation followed by rotation.

         Rotate the two adjacent lines by the same angle

         Join up the points