# 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 3  and the triangle has an area of 42 . What are the lengths of sides A and B?

May 4, 2017

If we presume that C is the base of the triangle:

A = 28.04
B = 28.04

#### Explanation:

First we need to find the height or $h$ of the triangle. We do this by using simple/basic algebra. In this case side C = $b$

= $\frac{b \cdot h}{2}$
= $\frac{3 \cdot h}{2} = 42$
= $\left(3 \cdot h\right) = 84$
= $h = \frac{84}{3}$
= $28$

Then we use one of the most powerful tools in geometry to decode the side lengths of A and B, the Pythagoras Theorem. In this case, as A and B meet at the vertex and a perpendicular drawn from this point on base bisects it. Hence, it forms a right angled triangle, whose two legs are $h = 28$ and $\frac{3}{2}$ (say $a$ and $b$) and one of the equal side forms hypotenuse or $c$. And then we have

${a}^{2} + {b}^{2} = {c}^{2}$
= ${\left(\frac{3}{2}\right)}^{2} + {28}^{2} = {c}^{2}$
= $2.25 + 784 = {c}^{2}$
= $786.25 = {c}^{2}$
= $\sqrt{786.25}$
$\approx$ $28.04$

All the best!