The hypotenuse is 10 and the area is 24, how do you find the other lengths on the triangle?

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The hypotenuse is 10 and the area is 24, how do you find the other lengths on the triangle?

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Answer 1 · 2015-12-29T11:51:57

$6$ $6$ and $8$ $8$ .

Explanation:

Let the other side lengths be $a$ $a$ and $b$ $b$ and area be $A$ $A$ . Then,
$\times A = 24$ $color(white)xxA=24$
$\Rightarrow \frac{a b}{2} = 24 \times \times \times \times \times \times \times \times$ $=>(ab)/2=24color(white)(xxxxxxxxxxxxxxxx)$ (area formula)
$\Rightarrow 2 a b = 96 \times \times \times \times \times \times \times \times$ $=>2ab=96color(white)(xxxxxxxxxxxxxxxx)$

$\Rightarrow a^{2} + b^{2} = 100 \times \times \times \times \times \times x$ $=>a^2+b^2=100color(white)(xxxxxxxxxxxxx)$ (The Pythagorean Theorem)

$\Rightarrow a^{2} + b^{2} + 2 a b = 100 + 2 a b \times \times \times \times \times \times x$ $=>a^2+b^2color(red)(+2ab)=100color(red)(+2ab)color(white)(xxxxxxxxxxxxx)$
$\Rightarrow {(a + b)}^{2} = 100 + 96 \times \times \times \times x$ $=>(a+b)^2=100+96color(white)(xxxxxxxxx)$ (A fundamental identity)
$\Rightarrow a + b = \sqrt{196} \times \times \times \times x$ $=>a+b=sqrt196color(white)(xxxxxxxxx)$
$\Rightarrow a + b = 14 \times \times \times \times x$ $=>a+b=14color(white)(xxxxxxxxx)$

The two number, of which product is $48$ $48$ and sum is $14$ $14$ , are $6$ $6$ and $8$ $8$ .
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The hypotenuse is 10 and the area is 24, how do you find the other lengths on the triangle?

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