Respuesta :

sqdancefan

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:

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,

[tex]\triangle ABC\sim \triangle BHC[/tex]

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

[tex]\implies \frac{BC}{HC}=\frac{AC}{BC}[/tex]

[tex]\frac{2}{HC}=\frac{AH+HC}{2}[/tex]

[tex]4=HC(HC+2+HC)[/tex]

[tex]4=HC(2HC+2)[/tex]

[tex]4=2HC(HC+1)[/tex]

[tex]2=HC(HC+1)[/tex]

[tex]\implies HC^2+HC-2=0[/tex]

[tex]HC^2+2HC-HC-2=0[/tex]

[tex]HC(HC+2)-1(HC+2)=0[/tex]

[tex](HC+2)(HC-1)=0[/tex]

By zero product property,

HC = -2 ( not possible ) or HC = 1  

Hence, the measure of HC is 1 unit.

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