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Similar triangles: matching sides in proportion

2026-09-18Plane GeometryMs Ho

Name corresponding vertices first, then set up a proportion with matching sides.

Key formulas

AB/DE = BC/EF = AC/DF
area ratio = (length ratio)²

Correspondence before algebra

If △ABC ∼ △DEF, then A maps to D, B to E and C to F. Matching sides are AB with DE, BC with EF, and AC with DF. A ratio written with unmatched sides is the most common error.

Worked example

△ABC ∼ △DEF with AB = 6, DE = 9 and BC = 10. Find EF. Because AB/DE = BC/EF, we have 6/9 = 10/EF. Cross-multiply: 6 × EF = 90, so EF = 15.

Area follow-up

The length ratio is 6:9 = 2:3, so the area ratio is 4:9. If the smaller triangle has area 20 cm², the larger has area 20 × (9/4) = 45 cm².

Worked example

Worked example

Problem. △ABC ∼ △DEF with AB = 6, DE = 9 and BC = 10. Find EF.

Solution. Scale factor from ABC to DEF is 9/6 = 3/2. Corresponding sides: EF = 10 × 3/2 = 15. So EF = 15.

Scale factor

Scale factor from ABC to DEF is 9/6 = 3/2. Multiply every length in ABC by 3/2 to get the matching length in DEF.

Review quiz

△PQR ∼ △STU with PQ=4, ST=14, QR=8. Then TU is:

Practice questions

  1. △PQR ∼ △XYZ with PQ = 8, XY = 12 and PR = 10. Find XZ.
  2. If the length ratio is 3:5, what is the area ratio?

Study notes

Does similarity mean the triangles are the same size?

No. Same shape, possibly different size. Corresponding angles are equal.

When do areas use the square of the ratio?

Always for similar plane figures: area ratio = (length ratio)².

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