Question

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

Answer 1

Answer:

  HC = 1

Step-by-step explanation:

AC = HC +AH = HC +(HC+2) = 2·HC +2

The altitude divides the triangle into similar triangles, so the ratio of hypotenuse to short side is the same for all. That is ...

  BC/HC = AC/BC

  2/HC = (2HC +2)/2

  4 = 2(HC)(HC +1) . . . . . cross multiply

  0 = HC² +HC -2 . . . . . . divide by 2, subtract 2

  0 = (HC -1)(HC +2) . . . . factor. Solutions are those values of HC that make the factors be zero.

The useful solution is ...

  HC = 1

Answer 2

Answer:

The measure of HC is 1 unit.

Step-by-step explanation:

Given,

In triangle ABC,

∠ABC = 90°, BC = 2,

Also, H∈ AC such that ∠BHC = 90°,

And, AH = HC + 2

We have to find : HC

∵ ∠ABC = ∠BHC ( right angles )

∠ACB = ∠HCB

By the AA similarity postulate,

$\triangle ABC\sim \triangle BHC$

∵ The corresponding sides of similar triangles are in same proportion,

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

$\frac{2}{HC}=\frac{AH+HC}{2}$

$4=HC(HC+2+HC)$

$4=HC(2HC+2)$

$4=2HC(HC+1)$

$2=HC(HC+1)$

$\implies HC^2+HC-2=0$

$HC^2+2HC-HC-2=0$

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

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

By zero product property,

HC = -2 ( not possible ) or HC = 1  

Hence, the measure of HC is 1 unit.