If the area of a right triangle is 15, what is its perimeter?

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If the area of a right triangle is 15, what is its perimeter?

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Answer 1 · 2018-04-23T02:51:35

The perimeter is given as a function of side $a$ $a$ via $p = a + \frac{30}{a} + \sqrt{a^{2} + \frac{30^{2}}{a^{2}}}$ $p = a + 30/a + sqrt{a^2 + 30^2/a^2}$ which has a minimum at $a = \sqrt{30}$ $a=sqrt{30}$ and is unbounded, no maximum.

Explanation:

It could be almost anything. Call the sides $a$ $a$ and $b$ $b$ and the hypotenuse $c$ $c$ . Of course

$c^{2} = a^{2} + b^{2}$ $c^2 =a^2 + b^2$

We know

$\frac{1}{2} a b = 15$ $1 /2 a b = 15$

$b = \frac{30}{a}$ $b = 30/a$

$c^{2} = a^{2} + {(\frac{30}{a})}^{2} = a^{2} + \frac{900}{a^{2}}$ $c^2 = a^2 + (30/a)^2 = a^2 + 900/a^2$

$c = \sqrt{a^{2} + \frac{900}{a^{2}}}$ $c = sqrt{ a^2 + 900/a^2 }$

Call the periimeter $p$ $p$ :

$p = a + b + c = a + \frac{30}{a} + \sqrt{a^{2} + \frac{900}{a^{2}}}$ $p = a+b+c = a + 30/a + sqrt{a^2 + 900/a^2}$

That's a function with a minimum at $a = \sqrt{30}$ $a=sqrt{ 30}$ (for positive $a$ $a$ ) and is unbounded, no maximum.

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If the area of a right triangle is 15, what is its perimeter?

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