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Find the missing length indicated.

 

 Mar 8, 2021
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5) The hypotenuse of the "12 by 16" triangle, using pythagoras' theorem, is \(\sqrt{12^2 + 16^2} = \sqrt{400} = 20\)

Then, the hypotenuse of the "12 by x" triangle, using the big triangle, is \(\sqrt{(x+16)^2 - 20^2}\)

But we also know that that side is \(\sqrt{12^2 + x^2}\)

So we can make an equation since the two are equivalent:
\(\sqrt{(x + 16)^2 - 20^2} = \sqrt{12^2 + x^2}\)

\((x + 16)^2 - 20^2 = 12^2 + x^2\)

\((x + 16)(x + 16) - x^2 = 144 + 400\)

\(x^2 - x^2 + 32x + 256 = 544\)

\(32x = 288\)

\(x = 9\)

 

6) This question is similar to the previous one so I won't explain in depth:
\(\sqrt{60^2 - 36^2} = 48\)

\(\sqrt{(x - 36)^2 + 48^2} = \sqrt{x^2 - 60^2}\)

\((x - 36)(x - 36) - x^2 = - 60^2 - 48^2\)

\(x^2 - x^2 - 72x + 1296 = -5904\)

\(-72x = -4608\)

\(x = 64\)

 Mar 8, 2021

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