ACT Math · Triangles and similarity

In a 30-60-90 triangle, the hypotenuse has length 14. Which value is closest to the length of the longer leg?

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In a 30-60-90 triangle, the hypotenuse has length 14. Which value is closest to the length of the longer leg?

Answer and explanation

Answer: B: 12.1

In a 3030-6060-9090 triangle, the side lengths are in the ratio 1:3:21:\sqrt3:2, with the hypotenuse the longest. The hypotenuse is 1414, so the shorter leg is 142=7\frac{14}{2}=7 and the longer leg is 73≈7(1.732)≈12.17\sqrt3\approx7(1.732)\approx12.1, choice B. Check with the Pythagorean theorem: 72+(73)2=49+147=196=1427^2+(7\sqrt3)^2=49+147=196=14^2. Choice A (77) is the shorter leg. Choice C (9.99.9) is 727\sqrt2, from a 4545-4545-9090 ratio. Choice D (1414) is the hypotenuse, which is always longer than either leg.

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