Chapter 12.6, Problem 22STP

To determine

To find: the base and the height of triangle

Answer to Problem 22STP

The length of base of triangle is 13 inches and height of triangle is 8 inches, hence the correct option is C

Explanation of Solution

Given information:

Area of triangle is 52 square inches.

Height of triangle is 5 inches less than length of its base.

Let length of the base of triangle be x inches.

Therefore height of the triangle is $\left(x-5\right)$ inches.

Area of the triangle is 52 square inches.

  $\begin{array}{l}\text{area of the triangle = }\frac{1}{2}\times \left(base\right)\times \left(height\right)\\ 52=\frac{1}{2}\times x\times \left(x-5\right)\left[\text{substitute the values of base and height}\right]\\ 52=\frac{x^2-5x}{2}\left[\text{multiply}\right]\\ 104=x^2-5x\left[\text{multiply both sides by 2}\right]\\ x^2-5x-104=0\left[\text{subtract 104 from both sides}\right]\\ x^2-13x+8x-104=0\left[-5x=-13x+8x\right]\\ x\left(x-13\right)+8(x-13)=0\\ \left(x-13\right)\left(x+8\right)=0\end{array}$

Either $x=13$ or $x=-8$ ,

  $x=13$ because length of base of triangle cannot be a negative number.

Height of triangle is $\left(x-5\right)$

  $\begin{array}{l}=13-5\left[\text{substitute the value of }x\right]\\ =8\end{array}$

Therefore,

The length of base of triangle is 13 inches and height of triangle is 8 inches.