Asked by joy
What are similar triangles?
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Answered by
GPT-5 mini
AI
Similar triangles are triangles that have the same shape but possibly different sizes.
Key points
- Corresponding angles are equal. If ΔABC is similar to ΔA'B'C' then ∠A = ∠A', ∠B = ∠B', ∠C = ∠C'.
- Corresponding sides are proportional. AB/A'B' = BC/B'C' = CA/C'A' = k (the scale factor).
Common tests for similarity
- AA (angle–angle): If two angles of one triangle equal two angles of another, the triangles are similar.
- SAS (side–angle–side) similarity: If two sides of one triangle are proportional to two sides of another and the included angles are equal, the triangles are similar.
- SSS (side–side–side) similarity: If all three pairs of corresponding sides are proportional, the triangles are similar.
Consequences
- Side lengths scale by k.
- Areas scale by k^2.
- Corresponding altitudes, medians and other corresponding segments are proportional to k.
Example
Δ with sides 3, 4, 5 is similar to Δ with sides 6, 8, 10 (scale factor k = 2). Their angles match and the area of the larger is 4 times the smaller (k^2 = 4).
Notation: ΔABC ~ ΔA'B'C' means triangle ABC is similar to triangle A'B'C'.
Key points
- Corresponding angles are equal. If ΔABC is similar to ΔA'B'C' then ∠A = ∠A', ∠B = ∠B', ∠C = ∠C'.
- Corresponding sides are proportional. AB/A'B' = BC/B'C' = CA/C'A' = k (the scale factor).
Common tests for similarity
- AA (angle–angle): If two angles of one triangle equal two angles of another, the triangles are similar.
- SAS (side–angle–side) similarity: If two sides of one triangle are proportional to two sides of another and the included angles are equal, the triangles are similar.
- SSS (side–side–side) similarity: If all three pairs of corresponding sides are proportional, the triangles are similar.
Consequences
- Side lengths scale by k.
- Areas scale by k^2.
- Corresponding altitudes, medians and other corresponding segments are proportional to k.
Example
Δ with sides 3, 4, 5 is similar to Δ with sides 6, 8, 10 (scale factor k = 2). Their angles match and the area of the larger is 4 times the smaller (k^2 = 4).
Notation: ΔABC ~ ΔA'B'C' means triangle ABC is similar to triangle A'B'C'.
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