Similarity and Congruence

  1

Two objects are similar if they have the same shape, so that one is an enlargement of the other.

Two objects are congruent if they are the same shape and size.

 

Scale Models and Factors

In an enlargement or reduction

 

Example

a c

Triangle A is an enlargement of C, with a scale factor of   2

Triangle C is a reduction of A, with a scale factor of   ½
      

2


Triangle B is an enlargement of C , with a scale factor of   3

Triangle B is an enlargement of A, with a scale factor of 1.5
Its  scale factor may be written as

1

Calculations with similar shapes

 

Two shapes are said to be similar if one is a scaled version of the other.
   This means that

Example 

Are these dominoes similar?

dom

      (All right angles, so equal)

2 and 3

 

The dominoes are similar

Missing Sides

Start with the shape containing the missing side.

 4

Example

Calculate the size of side x.

4

5

 

Similar Triangles

For any triangle
If pairs of corresponding angles are equal,
then ratios of lengths of corresponding sides are equal and the triangles are similar.

Conversely, if ratios of lengths of corresponding sides are equal, then pairs of corresponding angles are equal , and the triangles are similar.

 

Example

Given AD is 15 cm , AB is 12 cm and DE is 10cm,find the length of BC.

sim


BC is parallel to DE , so the corresponding angles are equal.

This means that triangles  ABC and ADE are similar.

Split the triangles into their components

1

2

1

6

 

 Scaled Area

12

7 8

9

 

Example

Given that shape A is enlarged by a scale factor of 2,

find the size of the Area of shape B.

12

112

10

 

Scaled Volume

 

simvol

 

12 13

10

 

Example

Wiffoff deodorant is usually sold in 200 ml cans.

A new size can is made, with half of the dimensions of the old one.

Calculate the volume of the new can.

pongo

 

14

The new can has a volume of 25 ml.

 

© Alexander Forrest