Question

In triangle △ABC, ∠ABC=90°, BH is an altitude. Find the missing lengths. AC=5 and BH=2, find AH and CH.

Answer 1

AH = 1 or 4

CH = 4 or 1

Explanation:

An altitude divides a right triangle into similar triangles. That means the sides are in proportion, so ...

AH/BH = BH/CH

AH·CH = BH²

The problem statement tells us AH + CH = AC = 5, so we can write

AH·(5 -AH) = BH²

AH·(5 -AH) = 2² = 4

This gives us the quadratic ...

AH² -5AH +4 = 0 . . . . in standard form

(AH -4)(AH -1) = 0 . . . . factored

This equation has solutions AH = 1 or 4, the values of AH that make the factors be zero. Then CH = 5-AH = 4 or 1.