# An isosceles triangle has sides A, B, and C, such that sides A and B have the same length. Side C has a length of 8  and the triangle has an area of 64 . What are the lengths of sides A and B?

Mar 8, 2018

Side color(brown)(a = b = 16.49 units

#### Explanation:

A median if triangle divides it in 2 triangles of equal areas. And if it is isosceles triangle the median is perpendicular bisector.

$c = 8$

$a = b , h e i g h t = h$

Half base = c / 2 = 8 / 2 = 4

Area of triangle = 64 sq units

Area of half triangle = 32 sq units

1/2 * 4 * h = 32

$h e i g h t$ $h = 16$ units

By using Pythagoras theorem

a = b = sqrt(4^2 + 16^2) = sqrt 272 = color(brown)(16.49 units

Mar 8, 2018

$A = B = 4 \sqrt{17} \text{units}$

#### Explanation:

Let $h$ the height $\text{area of triangle} = \frac{1}{2} \times h \times C$
$64 = \frac{1}{2} \times h \times 8$
$h = 16 \text{ units}$

∆ MHN: #
$M {N}^{2} = {h}^{2} + N {H}^{2}$
${A}^{2} = {16}^{2} + {\left(\frac{8}{2}\right)}^{2}$
$A = \sqrt{272}$
$A = 4 \sqrt{17}$

$A = B = 4 \sqrt{17} \text{units}$