# The height of a triangle is 5 m less than half its base. If the area of the triangle is 300 m2, how do you find the measure of the height?

Aug 18, 2016

height$= 15 \text{ meters}$

#### Explanation:

The formula for area of the triangle is $A = \frac{b h}{2}$.

Let the base be $b$ and the height be $\frac{b}{2} - 5$.

Then:

$300 = \frac{b \left(\frac{b}{2} - 5\right)}{2}$

$600 = b \left(\frac{b}{2} - 5\right)$

$600 = {b}^{2} / 2 - 5 b$

$600 = \frac{{b}^{2} - 10 b}{2}$

$1200 = {b}^{2} - 10 b$

${b}^{2} - 10 b - 1200 = 0$

Solve by completing the square:

$1 \left({b}^{2} - 10 b + 25 - 25\right) = 1200$

$1 \left({b}^{2} - 10 b + 25\right) - 25 = 1200$

${\left(b - 5\right)}^{2} = 1225$

$b - 5 = \pm 35$

$b = - 30 \mathmr{and} 40$

Hence, the base measures #40" meters" (a negative length is impossible).

The height therefore measures $\frac{40}{2} - 5 = \textcolor{g r e e n}{15}$

Hopefully this helps!