In triangle △ABC, ∠ABC=90°, BH is an altitude. Find the missing lengths. h AH=HC+2 and BC=2, find HC.

Question

In triangle △ABC, ∠ABC=90°, 

BH

 is an altitude. Find the missing lengths.  

Chapter Reference

h

AH=HC+2 and BC=2, find HC.

Answer 1

Let x = HC, then Drawing the altitude from B perpendicular to AC yields 3 similar triangles.

Then (2x + 2):2 = 2:x or (2x +2)/2 = 2/x

Cross multiplying yields: 2x² + 2x - 4 = 0 Factoring: (x + 2)(x - 1) = 0

x = -2 and x = 1

x cannot = - 2, so x=1

HC = 1