Similar Shapes

 

What Are Similar Shapes?

Two shapes are said to be similar when they:

• Have the same shape
• Have angles that match
• Have sides in proportion (i.e. linked by a scale factor)

One shape may be an enlargement or reduction of the other.

 

Proving That Shapes Are Similar

To show two non-triangular shapes are similar:

• Check that corresponding angles are equal
• Show that corresponding sides are in the same ratio

To show two triangles are similar:

• Prove all corresponding angles are equal
• Use angle facts: vertically opposite angles, alternate angles on parallel lines, angles in isosceles triangles, etc.

If all three corresponding angles are equal, the triangles are similar.

 

 

 

Example: Proving Rectangles Are Similar

(a) Prove the two rectangles below are similar

• Rectangle A: Length = 15 cm, Width = 5 cm
• Rectangle B: Length = 6 cm, Width = 2 cm

 

 

Step 1: Find the scale factor of length:

$$\frac{15}{6} = 2.5$$

Step 2: Find the scale factor of width:

$$\frac{5}{2} = 2.5$$

As both pairs of sides have the same scale factor. The rectangles are similar.

 

Example: Proving Triangles Are Similar

(b) Given AB and CD are parallel, prove that triangles $\triangle ABY$ and $\triangle CDY$ are similar.

 

 

Step 1: Identify equal angles:

• $\angle AYB = \angle CYD$ (Vertically opposite angles)
• $\angle ABY = \angle CDY$ (Alternate angles on parallel lines)
• $\angle BAY = \angle DCY$ (Alternate angles on parallel lines)

All corresponding angles are equal. Therefore, the triangles are similar.

 

Solving Problems With Similar Shapes

If shapes are similar, their sides are linked by a scale factor.

To find an unknown side:

1. Identify corresponding sides
2. Calculate the scale factor: $$\text{Scale Factor} = \frac{\text{Side on Shape B}}{\text{Corresponding Side on Shape A}}$$
3. Multiply/divide to find the unknown length.

 

Example: Similar Quadrilaterals

ABCD and PQRS are similar.

AB = 6 cm, PQ = 3 cm, AD = 15 cm. Find PS.

 

 

Step 1: Find the scale factor: $\text{Scale Factor} = \frac{3}{6} = 0.5$

Step 2: Use scale factor to find PS: $PS = 0.5 \times 15 = 7.5 \, \text{cm}$

$PS$ = 7.5 cm