Question

In triangle △ABC, ∠ABC=90°, BH is an altitude. Find the missing lengths.

Answer 1

Answer:

HC=$\sqrt{3}$

Step-by-step explanation:

Let HC=x, it is given that AH=3HC, then AH=3x.

Since, from the given figure, ΔABC is similar to ΔBHC and ΔABC is similar to ΔABH.

Therefore, ΔABH is similar to ΔBHC, hence using the similarity conditions,

$\frac{HC}{BH}=\frac{BH}{AH}$

$\frac{x}{3}=\frac{3}{3x}$

$3x^{2}=9$

$x^{2}=3$

$x=\sqrt{3}$

Hence, HC=$\sqrt{3}$.