Question

In triangle △ABC, ∠ABC=90°, BH is an altitude. Find the missing lengths. h AH=HC+2 and BC=2, find HC.
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Answer 1

Answer:

The value of HC is 1 unit.

Step-by-step explanation:

Given information: ∠ABC=90°, BH is an altitude and AH=HC+2.

Let the measure of HC be x.

$AH=HC+2=x+2$

In triangle ABC and triangle BHC,

$\angle B=\angle H=90^{\circ}$

$\angle C=\angle C$                         (Common angle)

By AA property of similarity,

$\triangle ABC\sim \angle BHC$  

The corresponding sides of a similar triangle are proportional.

$\frac{BC}{HC}=\frac{AC}{BC}$

$\frac{2}{x}=\frac{x+2+x}{2}$

$4=x(2x+2)$

$4=2x^2+2x$

$2x^2+2x-4=0$

$2(x^2+x-2)=0$

$x^2+2x-x-2=0$

$x(x+2)-(x+2)=0$

$(x+2)(x-1)=0$

Equate each factor equal to 0.

$x=-2$

$x=1$

The length can not be negative therefore value of HC is 1 unit.