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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+1280 
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First, we must set some variables to complete this problem. Let's set HB as x. 

Note that \(AC^2 = AB^2 - BC^2 = (2 +x)^2 - 1^2 = x^2 + 4x + 3\)

 

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

\(CH^2 = BC^2 - BH^2 = 1 - x^2 \\ CH^2 = AC^2 - AH^2 = (x^2 + 4x + 3) - 2^2 = x^2 + 4x -1 \)

 

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

\(1-x^2 = x^2 + 4x -1 \\ 2x^2 + 4x -2 = 0 \\ x^2 + 2x - 1 = 0 \\ x^2 + 2x = 1\)

\(x^2 + 2x + 1 = 1 + 1 \\ (x + 1)^2 = 2\\ x + 1 =\sqrt{2} \\ x =\sqrt{2} - 1 \)

 

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

\(CH^2 = BC^2 - BH^2 = 1^2 - (\sqrt{ 2} -1)^2 = 1 - (2 - 2\sqrt{ 2} + 1) = 2\sqrt {2} - 2 \\ CH = \sqrt {2\sqrt {2} - 2} = \sqrt{\sqrt {8} - 2} ≈ .91\)

 

So our final answer is about .91. 

 

Thanks! :)

 Jun 12, 2024

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