Overview

Similarity converts hard length problems into proportions. Once triangles are similar, everything scales predictably and angles line up immediately.

Key Ideas

• Similarity tests: AA, SAS, SSS.
• Corresponding sides are in a constant ratio $k$.
• Areas scale by $k^2$.
• Perimeters scale by $k$.

Core Skills

Match Corresponding Angles

Write angle correspondence first. This prevents mixing side ratios later.

Use Scale Factors

Once $k$ is found, every length, height, and perimeter scales by $k$. Areas scale by $k^2$.

Identify Similarity in Diagrams

Parallel lines create equal angles; right angles plus one acute angle also force AA similarity.

Worked Example

Two similar triangles have scale factor $k=2$. If the smaller area is $12$, find the larger area.

Area scales by $k^2=4$, so the larger area is $48$.

More Examples

Example 1: Side Lengths

Two triangles are similar with $k=3$. A side of the smaller triangle is $5$. Find the corresponding side in the larger triangle.

$5\cdot 3 = 15$.

Example 2: Perimeter

The smaller triangle has perimeter $18$ and $k=\tfrac{5}{2}$. Find the larger perimeter.

$18\cdot \tfrac{5}{2} = 45$.

Example 3: Area Ratio to Scale Factor

If the area ratio is $9:25$, what is the side-length ratio?

The scale factor is $3:5$.

Strategy Checklist

• Mark corresponding angles before writing any ratios.
• Use one pair of sides to find $k$.
• Apply $k$ to lengths and $k^2$ to areas.
• Check that proportional sides match the chosen correspondence.

Common Pitfalls

• Using perimeter ratios when area ratios are needed.
• Matching the wrong corresponding sides.
• Forgetting that similarity preserves angle measures exactly.

Practice Problems