Question

If a triangle that has an area of 110 square feet has a base that is two feet less than twice the height, what is the

length of its base and height?

Short answer

The height of the triangle is 11 feet and the base is 20 feet.

Step-by-step solution

Step 1. Given information.

Base of a triangle is two feet less than twice the height.

The area of the triangle is 110 square feet.

Step 2. Form a quadratic equation.

The area of a triangle is:

$A=\frac{1}{2}bh$

Where, $b$is the base of the triangle and $h$is the height of the triangle.

Using the given information, it is clear that $b=2h-2$.

Substitute $b=2h-2$and $A=110$in the above formula.

$\begin{array}{rcl}110& =& \frac{1}{2}(2h-2)h\\ 0& =& h^2-h-110\end{array}$

Step 3. Determine the height.

Split the middle term.

$\begin{array}{rcl}{h}^2-11h+10h-110& =& 0\\ h(h-11)+10(h+-11)& =& 0\\ (h+10)(h-11)& =& 0\\ h& =& -10\text{and}h=11\end{array}$

Height of a triangle cannot be negative. So, $h=11$.

Step 4. Determine the base.

The base of the triangle is:

$\begin{array}{rcl}b& =& 2h-2\\ & =& 2\left(11\right)-2\\ & =& 22-2\\ & =& 20\end{array}$

Step 5. Conclusion.

The height of the triangle is 11 feet and the base is 20 feet.