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# right triangle

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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