Question

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

Answer 1

Answer:

HC=1

Step-by-step explanation:

In triangle △ABC, ∠ABC=90°, BH is an altitude.

Please see the attachment for diagram.

In ΔABH and Δ BCH

∠ABH=∠BCH

∠AHB=∠BHC     (each 90°)

∴ ΔABH ≈ Δ BCH    By AA similarity

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

$BH^2=AH\times HC$

$2^2-HC^2=(HC+2)\times HC$

$4-HC^2=HC^2+2HC$

$HC^2+HC-2=0$

(HC+2)(HC-1)=0

HC=1 and -2

HC can't be negative.

Hence, HC=1