Triangles • Topic 3 of 5

Similar Triangles

Two triangles are similar when they have the same shape but not necessarily the same size: corresponding angles are equal and corresponding sides are in the same ratio, called the scale factor. If one triangle is a 3× enlargement of another, every side is three times as long. To find a missing side, set the matching side ratios equal and solve the proportion. Similar triangles appear in shadow, mirror and nested-triangle problems and in any figure with parallel lines cutting a triangle. The SAT often gives one pair of corresponding sides to fix the scale factor, then asks for another side using that ratio.

A small 3-4-5 right triangle and a similar 6-8-10 triangle with scale factor 2Similar triangles3456810Scale factor 2: corresponding sides in the same ratio

✅ Solved examples

1. Triangles are similar; a side of 4 corresponds to 12. What is the scale factor?
12 ÷ 4 = 3.
2. With that scale factor, a side of 5 corresponds to what length?
5 × 3 = 15.
3. Similar triangles: 6 corresponds to 18, and 8 corresponds to x. Find x.
Scale 3, so x = 8 × 3 = 24.
4. Are all equilateral triangles similar?
Yes — their angles are all 60°, so shapes match.

✏️ Practice — try these, take hints as needed

1. A side of 5 corresponds to 20. What is the scale factor?
Divide corresponding sides.
20 ÷ 5.
—
4.
2. Scale factor 4: a side of 3 corresponds to what length?
Multiply by 4.
3 × 4.
—
12.
3. Similar triangles: 4→8 and 7→x. Find x.
Scale = 8 ÷ 4 = 2.
7 × 2.
—
14.
4. Similar triangles: 10→25 and 6→x. Find x.
Scale = 25 ÷ 10 = 2.5.
6 × 2.5.
—
15.
5. In similar triangles, corresponding angles are:
Same shape.
—
—
Equal.

📝 Topic test — 8 questions

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