Maths Mutt HOME

Similarity and Congruence

Similarity diagram

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 have the same shape and size.

Scale Models and Factors

In an enlargement or reduction:

  • Corresponding angles are equal
  • Corresponding lengths are in the same ratio (same scale factor)
  • Scale factor > 1 → enlargement
  • Scale factor < 1 → reduction
Example
Triangle A Triangle C

Triangle A is an enlargement of C with scale factor 2.

Triangle C is a reduction of A with scale factor \( \tfrac12 \).

Triangle B

Triangle B is an enlargement of C with scale factor 3.

Triangle B is an enlargement of A with scale factor 1.5.

Scale factor equation

Calculations with Similar Shapes

Two shapes are similar if one is a scaled version of the other. This means:

  • Corresponding angles are equal
  • Corresponding sides are in the same ratio (same scale factor)
Example

Are these dominoes similar?

Dominoes
  • All angles are right angles → equal
  • Check ratios of corresponding sides
Ratio 1 Ratio 2

The dominoes are similar.

Missing Sides

Start with the shape containing the missing side.

$$ x = (\text{scale factor}) \times (\text{corresponding side}) $$
Example

Calculate the size of side x.

Missing side diagram
$$ \text{scale factor} = \frac{5}{20} = \frac14 $$ $$ x = \text{scale factor} \times \text{corresponding side} $$ $$ x = \frac14 \times 12 $$ $$ x = 3 $$

Similar Triangles

For any triangle: If pairs of corresponding angles are equal, then ratios of corresponding sides are equal → triangles are similar.

Conversely, if ratios of corresponding sides are equal, then corresponding angles are equal → triangles are similar.

Example

Given AD = 15 cm, AB = 12 cm, DE = 10 cm, find BC.

Triangle similarity

BC is parallel to DE → corresponding angles equal → triangles ABC and ADE are similar.

Split the triangles into components:

Triangle components 1 Triangle components 2
$$ x = (\text{scale factor}) \times (\text{corresponding side}) $$
$$ \text{scale factor} = \frac{AB}{AD} = \frac{12}{15} = \frac45 $$ $$ \text{corresponding side} = DE = 10\text{ cm} $$ $$ x = \frac45 \times 10 $$ $$ x = \frac{40}{5} $$ $$ x = 8\text{ cm} $$

Scaled Area

Scaled area
$$ \text{Area } A = 6\text{ cm}^2 $$ $$ \text{Scale factor} = 2 $$ $$ (\text{Scale factor})^2 = 4 $$
$$ \text{Area } B = 24\text{ cm}^2 $$ $$ \text{Area } B = 4 \times 6 $$ $$ = 4 \times \text{Area } A $$ $$ \text{Area } B = (\text{scale factor})^2 \times \text{original area} $$
$$ \text{Scaled Area} = (\text{scale factor})^2 \times \text{original area} $$
Example

Shape A is enlarged by scale factor 2. Find the area of shape B.

Area example Area working
$$ \text{Area } B = (\text{scale factor})^2 \times \text{original area} $$ $$ = 2^2 \times 16 $$ $$ = 4 \times 16 $$ $$ = 64\text{ cm}^2 $$

Scaled Volume

Scaled volume
$$ \text{Area } A = 6\text{ cm}^2 $$ $$ \text{Volume } A = 6\text{ cm}^3 $$ $$ \text{Scale factor} = 2 $$ $$ (\text{Scale factor})^3 = 8 $$
$$ \text{Area } B = 24\text{ cm}^2 $$ $$ \text{Volume } B = 48\text{ cm}^3 $$ $$ \text{Volume } B = 8 \times 6 $$ $$ = 8 \times \text{Volume } A $$
$$ \text{Scaled Volume} = (\text{scale factor})^3 \times \text{original volume} $$
Example

Wiffoff deodorant is sold in 200 ml cans. A new can has half the dimensions. Find its volume.

Deodorant can
$$ \text{New Volume} = (\text{scale factor})^3 \times \text{original volume} $$ $$ = \left(\frac12\right)^3 \times 200 $$ $$ = \frac18 \times 200 $$ $$ = 25\text{ ml} $$

The new can has a volume of 25 ml.