Congruence and Similarity

What you will learn

• Say when two shapes are congruent and when they are similar.
• Find a missing length using a scale factor.
• Relate scale factors for length and area.

Before you start: you should know the four transformations. See Transformations.

The idea

Two shapes are congruent if one fits exactly onto the other after sliding, flipping or turning it. They have the same size and the same shape. Two shapes are similar if one is an enlargement of the other. They have the same shape but not always the same size.

Similar shapes and the scale factor

In similar shapes all matching angles are equal and all matching lengths are in the same ratio. That ratio is the scale factor $k$.

To find a missing side, first find $k$ from a pair of matching sides you know, then multiply.

Lengths and areas

Lengths multiply by $k$, but areas multiply by $k^2$. Double every length of a rectangle and its area is four times bigger.

Proving triangles congruent

Two triangles are congruent if you can show any one of these: SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and a matching side) or RHS (a right angle, the hypotenuse and one other side). Three equal angles (AAA) only prove similarity.

Worked examples

Example 1: Missing side

Triangles $ABC$ and $DEF$ are similar. $AB = 4$ cm matches $DE = 10$ cm. $BC = 6$ cm. Find $EF$.

1. $k = 10 \div 4 = 2.5$.
2. $EF = 6 \times 2.5 = 15$.

Answer: $15$ cm

Example 2: Area

A photo $10$ cm wide is printed $30$ cm wide. The original area is $60 \text{ cm}^2$. Find the new area.

1. $k = 30 \div 10 = 3$.
2. Area factor $= 3^2 = 9$.
3. $60 \times 9 = 540$.

Answer: $540 \text{ cm}^2$

Example 3: Scale drawing

A plan uses $1$ cm for every $2$ m. A room is $7.5$ cm long on the plan. How long is the real room?

1. The scale is $1$ cm to $2$ m, so $7.5 \times 2 = 15$.

Answer: $15$ m

Tip: write the scale factor as “new over old”. Then multiply to enlarge and divide to go back.

Why it matters

Maps, plans, models and photographs are all similar to the real thing. Architects and model makers use scale factors every day, and the idea that area grows with the square of the scale explains why a giant could not be a simple enlargement of a person.