Similarity (With Applications to Maps and Models)
313. The Shape-Shifting Rule · Equal angles, proportional sides
Corresponding angles are equal and corresponding sides are proportional.
For similar triangles, corresponding angles are equal and corresponding sides are proportional (a constant ratio). Either condition, correctly matched, brings the other with it.
What this lesson covers
Try to break it
Drag A, B, or C to reshape △ABC. △PQR scales and shifts to keep its angles equal to ABC's. Once the angles match, the side ratios are automatically fixed — try to force one ratio to be different from the others; impossible. Same angles ⇒ same shape (up to scale).
How you build it
Construct a triangle similar to ABC.
- A is at (2, 2). For a half-size copy, plot P halfway from O to A along ray OA — at (1, 1).
- B is at (8, 2). Plot Q halfway from O to B along ray OB — at (4, 1).
- C is at (4, 6). Plot R halfway from O to C along ray OC — at (2, 3).
- Join P to Q. PQ corresponds to AB, and PQ = ½ AB.
- Join Q to R. QR corresponds to BC, and QR = ½ BC.
- Join R to P. Triangle PQR has the same angles as ABC and every side half as long, so AB/PQ = BC/QR = CA/RP = 2 — the triangles are similar.
The proof, step by step
Prove that the triangles are similar — equal angles and proportional sides.
- By construction, vector PQ is a scaled version of AB (PQ = k·AB), and vector PR is a scaled version of AC (PR = k·AC).
- Uniform scaling preserves angles. Therefore, the angle between PQ and PR equals the angle between AB and AC, so ∠P = ∠A.
- Similarly, ∠Q = ∠B and ∠R = ∠C. All corresponding angles are equal.
- The sides are proportional by definition: AB/PQ = BC/QR = AC/PR = 1/k. Hence, ΔABC ~ ΔPQR.
Worked example
In ΔABC and ΔPQR, ∠A = ∠P = 60°, ∠B = ∠Q = 70°. If AB = 6 cm, PQ = 4 cm, and BC = 9 cm, find the length of QR.
Since ΔABC ~ ΔPQR, corresponding sides are proportional: AB/PQ = BC/QR. Substituting values: 6/4 = 9/QR ⇒ QR = (9 × 4)/6 = 6 cm.
- 6 cm — correct
- 4.5 cm
- 13.5 cm
- 3 cm