# If a right triangle has a hypotenuse of 6, and a perimeter of 14, what is the area of the right triangle?

Apr 25, 2018

$\textcolor{b l u e}{6 \setminus \setminus \setminus \setminus {\text{units}}^{2}}$

#### Explanation:

By Pythagoras' theorem, the square on the hypotenuse is equal to the sum of the squares of the other two sides.

Let the two unknown sides be $\boldsymbol{a} \mathmr{and} \boldsymbol{b}$

Then:

${a}^{2} + {b}^{2} = 36 \setminus \setminus \setminus \setminus \setminus \left[1\right]$

The perimeter of a triangle is the sum of all its sides.

$a + b + 6 = 14 \setminus \setminus \setminus \setminus \setminus \left[2\right]$

Using $\left[2\right]$

$a = 6 - b$

Substituting in $\left[1\right]$

${\left(6 - b\right)}^{2} + {b}^{2} = 36$

Expanding:

${b}^{2} - 12 b + 36 + {b}^{2} = 36$

$2 {b}^{2} - 12 b = 0$

Factor:

$2 b \left(b - 6\right) = 0 \implies b = 0 \mathmr{and} b = 6$

$b = 0$ is not valid, we can't have a triangle with no side.

Substituting $b = 6$ in $\left[2\right]$

$a + 6 + 6 = 14$

$a = 2$

Area of triangle is:

$\frac{1}{2} \text{base"xx"height}$

1/2(6)*(2)=6 \ \ \ \"units^2