# 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 9  and the triangle has an area of 90 . What are the lengths of sides A and B?

Jun 4, 2016

Lengths of side A and B are $20.5$ units.

#### Explanation:

Let the sides A and B be $a$ units each. As side C has a length of $9$ units, one can draw a perpendicular from point opposite which divides the side C in two equal parts.

Hence, height of perpendicular using Pythagoras theorem will be

$\sqrt{{a}^{2} - {\left(\frac{9}{2}\right)}^{2}}$ and as area of triangle is $90$ units

$\sqrt{{a}^{2} - {\left(\frac{9}{2}\right)}^{2}} \times \frac{9}{2} = 90$

Hence $\sqrt{{a}^{2} - {\left(\frac{9}{2}\right)}^{2}} = 90 \times \frac{2}{9} = 20$

or ${a}^{2} - {\left(\frac{9}{2}\right)}^{2} = 400$

${a}^{2} = 400 + \frac{81}{4} = \frac{1681}{4}$

or $a = \frac{41}{2} = 20.5$

Lengths of side A and B are $20.5$ units.