Geometry and Trigonometry practice question

Answer: C. 270

Since sin A = opposite/hypotenuse = BC/AB = 5/13, triangle ABC is similar to a 5-12-13 right triangle, so the sides are 5k, 12k, and 13k for some k > 0. The perimeter gives 5k + 12k + 13k = 30k = 90, so k = 3 and the sides are 15, 36, and 39. The legs are 15 and 36, so the area = (1/2)(15)(36) = 270. (A) is the area of the base 5-12-13 triangle, using the ratio's numbers as actual side lengths and ignoring the given perimeter. (B) comes from scaling the area of the 5-12-13 triangle by the linear factor 3 instead of the square of the factor, 3² = 9 (30 · 3 = 90 rather than 30 · 9 = 270). (D) omits the factor of 1/2, computing 15 · 36 = 540, the area of the corresponding rectangle.

Why the other answers are wrong

A. 30

You used 5, 12, and 13 as the actual side lengths, but those give a perimeter of 30, not 90. Scale by k = 3 first to get legs 15 and 36, so the area is 270.

B. 90

You scaled the area 30 by the linear factor 3, but areas scale by the square of the factor: 30 · 3² = 270.

D. 540

15 · 36 = 540 is the product of the legs — the area of the corresponding rectangle. A triangle needs the factor of ½: (1/2)(15)(36) = 270.

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