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

Dec 29, 2015

$6$ and $8$.

#### Explanation:

Let the other side lengths be $a$ and $b$ and area be $A$. Then,
$\textcolor{w h i t e}{\times} A = 24$
$\implies \frac{a b}{2} = 24 \textcolor{w h i t e}{\times \times \times \times \times \times \times \times}$(area formula)
$\implies 2 a b = 96 \textcolor{w h i t e}{\times \times \times \times \times \times \times \times}$

$\implies {a}^{2} + {b}^{2} = 100 \textcolor{w h i t e}{\times \times \times \times \times \times x}$(The Pythagorean Theorem)

$\implies {a}^{2} + {b}^{2} \textcolor{red}{+ 2 a b} = 100 \textcolor{red}{+ 2 a b} \textcolor{w h i t e}{\times \times \times \times \times \times x}$
$\implies {\left(a + b\right)}^{2} = 100 + 96 \textcolor{w h i t e}{\times \times \times \times x}$(A fundamental identity)
$\implies a + b = \sqrt{196} \textcolor{w h i t e}{\times \times \times \times x}$
$\implies a + b = 14 \textcolor{w h i t e}{\times \times \times \times x}$

The two number, of which product is $48$ and sum is $14$, are $6$ and $8$.