Question

Given: △ABC, m∠B=90° AB=12, BC=16, BK ⊥ AC . Find: AC and BK. Please explain how to find out BK

Answer 1

AC=20, BK=9.6

Explanation:

Answer 2

• AC = 20
• BK = 9.6

Explanation:

AC is found using the Pythagorean theorem in the usual way:

AC² = AB² + BC²

AC² = 12² + 16² = 144 +256 = 400

AC = √400

AC = 20

You may recognize this is a 3-4-5 triangle scaled by a factor of 4.

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Then BK is found using similar triangles. All of the triangles in the figure are similar*, so ...

long side / hypotenuse = BC/AC = BK/AB

BK = AB(BC/AC) = 12(16/20) = 12×0.8

BK = 9.6

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* You can take it on faith, or you can prove it. The basic idea is that all of the triangles are right triangles and any two have at least one pair of corresponding acute angles that are congruent. Hence, they are AA similar. ∠ABK and ∠CBK are complementary, as are angles A and C. Of course ∠A is complementary to ∠ABK. Any angles complementary to the same angle are congruent.