Similar Shapes and Scale Factors

Grade 5

What makes shapes similar?

Two shapes are similar if they have the same shape but different sizes — all corresponding angles are equal and all corresponding sides are in the same ratio. That ratio is the scale factor.

Similar shapes are like enlargements of each other. Congruent shapes are a special case where the scale factor is 1.

Finding missing lengths

Find the scale factor by dividing a known side of the larger shape by the corresponding side of the smaller shape. Then multiply (or divide) the other sides by the same factor.

Always match corresponding sides — sides opposite equal angles. Writing the sides in a ratio table helps: small shape on top, large on bottom, same position in each column.

Worked example

EXAMPLE

Two similar triangles have corresponding sides 4 cm and 10 cm. The smaller triangle has a second side of 6 cm. Find the corresponding side in the larger triangle.

Scale factor = 10 ÷ 4 = 2.5. Missing side = 6 × 2.5 = 15 cm.

Try this one

\( ABCD \) is a parallelogram. The diagonal \( AC \) is drawn, splitting it into triangle \( ABC \) and triangle \( CDA \).

Prove that triangle \( ABC \) is congruent to triangle \( CDA \). State the congruence condition you use. [3]

Nova's hint:

Use the properties of a parallelogram, together with the diagonal, to match up parts of the two triangles.

Ask yourself: 'which pairs of sides can I show are equal, and is there a side that both triangles share?' Then decide which congruence condition those facts satisfy.

Try to Solve it with Nova?