# 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 2  and the triangle has an area of 16 . What are the lengths of sides A and B?

Find the height of the triangle by cutting it in half and making a right triangle, then use the pythagorean theorem to get to sides A and B with lenth
$\sqrt{65}$

#### Explanation:

We have an isosceles triangle with base of 2 and sides A and B. The whole triangle has area of 16. What is the length of a side?

$A = b h$

Let's think about this triangle for a second - what we're trying to find is the length of a side, so either A or B will do. So we don't need to work with the whole triangle. Instead, we really only have to work with 1/2 of the triangle - that would create a right triangle where one base is the height of the whole triangle, another base is 1/2 the base of the whole triangle (and so that measure is 1) and half the area (which would be 8).

We can now figure out the height:

$8 = 1 \left(h\right)$

$h = 8$

And with that, we can now figure out the sloping side of the whole triangle, which is also the hypotenuse of the right triangle, using the pythagorean theorem:

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

${1}^{2} + {8}^{2} = {c}^{2}$

$1 + 64 = {c}^{2}$

$c = \sqrt{65}$

And so sides A and B are length $\sqrt{65}$