# In a trapezoid, the smaller base is 3 more than the height, the larger base is 5 less than 3 times the height, and the area of the trapezoid 45 square centimeters. How do you find, in centimeters, the height of the trapezoid?

Nov 21, 2016

The height is $5$ cm.

#### Explanation:

Let the height be $x$, the smaller base be $x + 3$ and the larger base be $3 x - 5$. Recall the area of a trapezoid is given by $A = \frac{\left({b}_{1} + {b}_{2}\right) h}{2}$, where ${b}_{1}$ and ${b}_{2}$ are the two bases.

$45 = \frac{\left(x + 3 + 3 x - 5\right) x}{2}$

$90 = \left(4 x - 2\right) x$

$90 = 4 {x}^{2} - 2 x$

$0 = 4 {x}^{2} - 2 x - 90$

$0 = 2 \left(2 {x}^{2} - x - 45\right)$

$0 = 2 {x}^{2} - 10 x + 9 x - 45$

$0 = 2 x \left(x - 5\right) + 9 \left(x - 5\right)$

$0 = \left(2 x + 9\right) \left(x - 5\right)$

$x = - \frac{9}{2} \mathmr{and} 5$

A negative answer is impossible, so the height of the trapezoid is $5$ cm.

Hopefully this helps!