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Triangle

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A right triangle has legs of length 6 and b, and a hypotenuse of length c. The perimeter of the triangle is 15. Compute c.

May 15, 2022

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By Pythagorean theorem, $$6^2 + b^2 = c^2$$.

Perimeter is 15 means $$6 + b + c = 15$$.

Then $$\begin{cases}c^2 - b^2 = 36\\b + c = 9\end{cases}$$. Note that c^2 - b^2 = (c - b)(b + c) = 9(c - b). Therefore:

$$9(c - b) = 36\\ c - b = 4$$

Consider $$2c = (b + c) + (c - b)$$.

$$2c = 9 + 4\\ c = \dfrac{13}2$$

May 15, 2022