The right- triangle shown above, ( ), has , and . Find the altitude (height) from to with foot at , the lenght of side and .
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∠ A B C = x ⟹ ∠ A C B = 9 0 − x In Δ A B D ∠ B A D = 9 0 − x = ∠ A C B In Δ A C D ∠ C A D = x = ∠ A B C In Δ A B C and Δ D B A ∠ B A C = ∠ A D B = 9 0 ∠ B A D = ∠ A C B ⟹ Δ A B C Δ D B A (AA) ⟹ B D A B = A B B C ⟹ A B 2 = B C ⋅ A B ⟹ 1 5 2 = n ( n + 1 6 ) ⟹ 2 2 5 = n ( n + 1 6 ) ⟹ n = 9 , − 2 5 ⟹ n = 9 In Δ A B C B C 2 = A B 2 + A C 2 ⟹ b = A C 2 = B C 2 − A B 2 = 4 0 0 ⟹ b = 2 0 In Δ A D B Δ A B C Δ D B A ⟹ D B A B = A D A C ⟹ 9 1 5 = h 2 0 ⟹ h = 1 2 ∴ n + b + h = 1 2 + 2 0 + 9 = 4 1