Similarity (With Applications to Maps and Models)
312. Same Shape, Different Size · Exploring the definition of similar figures
The ratio of corresponding sides remains constant (equal to k) as you reshape the triangle.
Two figures are similar when they have the same shape but not necessarily the same size: their corresponding angles are equal and corresponding sides are in the same ratio k, the scale factor. Congruent figures are the special case k = 1.
What this lesson covers
Try to break it
Drag A to reshape △ABC, and drag the k slider to set the scale factor. △A'B'C' is always a scaled copy: corresponding angles match exactly, and every side is k times the corresponding side. The side ratio always equals k — try to drag in a way that breaks it; impossible.
How you build it
Construct a scaled copy of triangle ABC.
- A is at (1, 1). For k = 2, plot A' twice as far from O along ray OA — at (2, 2).
- B is at (3, 1). Plot B' twice as far from O along ray OB — at (6, 2).
- C is at (2, 3). Plot C' twice as far from O along ray OC — at (4, 6).
- Join A' to B' — the side matching AB, now twice as long.
- Join B' to C' — the side matching BC, twice as long.
- Join C' to A'. The triangle A'B'C' has the same angles as ABC and every side twice as long — same shape, double the size (k = 2).
The proof, step by step
Prove that the two triangles are similar — corresponding sides in a constant ratio.
- By construction, A'B'C' is a scaled copy of ABC from center O with factor k.
- Scaling preserves angles. Therefore, ∠A = ∠A', ∠B = ∠B', and ∠C = ∠C'.
- Each side length is multiplied by k. Thus, AB/A'B' = BC/B'C' = AC/A'C' = 1/k.
- Since corresponding angles are equal and corresponding sides are proportional, △ABC ~ △A'B'C'.
Worked example
△ABC ~ △DEF. If AB = 4 cm, DE = 6 cm, and the area of △ABC is 16 cm², find the area of △DEF.
Ratio of sides = 4/6 = 2/3. Ratio of areas = (2/3)² = 4/9. Area of △DEF = 16 × (9/4) = 36 cm².
- 24 cm²
- 32 cm²
- 36 cm² — correct
- 48 cm²