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In triangle ABC, angle ACB = 90 degrees. Let H be the foot of the altitude from C to side AB.
If BC = 1 and AH = 2, find CH.

 Jun 12, 2024
 #1
avatar+1952 
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First, we must set some variables to complete this problem. Let's set HB as x. 

Note that AC2=AB2BC2=(2+x)212=x2+4x+3

 

We can write a system of equations to help us solve this question. We have

CH2=BC2BH2=1x2CH2=AC2AH2=(x2+4x+3)22=x2+4x1

 

Since both equations are equal to CH^2, we set both of them to equal each other, and we get

1x2=x2+4x12x2+4x2=0x2+2x1=0x2+2x=1

x2+2x+1=1+1(x+1)2=2x+1=2x=21

 

Now, we move on to the final step, which is to find CH. Plugging x back into the original formula, we get

CH2=BC2BH2=12(21)2=1(222+1)=222CH=222=82.91

 

So our final answer is about .91. 

 

Thanks! :)

 Jun 12, 2024

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