Q36. - Algebra 1

Question

Write an algebraic expression to represent the area of the triangle. Then evaluate it to find the area when $h=12$ inches

Step-by-Step Solution

Answer

The algebraic expression for the area is $\frac{h^2}{2}+3h$ and when $h=12\text{\hspace{0.17em}inches}$, the area is $A=108{\text{\hspace{0.17em}inches}}^{\text{2}}$.

Step 1. Apply the formula.

The formula for the area of a triangle given by

$\frac{1}{2}\times \text{base×height}$

Step 2. Example of finding the area of a triangle.

If the height of a triangle is $3h+2$ and the base of the triangle is $\frac{2}{h}+1$ then the area is given by $\frac{1}{2}\times \text{base×height\hspace{0.17em}=\hspace{0.17em}}\frac{1}{2}\times (3h+2)\times \left(\frac{2}{h}+1\right)$.

When $h=2\text{\hspace{0.17em}inches}$, the area of the triangle is given by

$\begin{array}{l}=\frac{1}{2}\times \left(\right(3\times 2)+2)\times \left(\frac{2}{2}+1\right)\\ =\frac{1}{2}\times (6+2)\times (1+1)\\ =\frac{1}{2}\times 8\times 2\\ =8{\text{\hspace{0.17em}inches}}^{\text{2}}\end{array}$

Step 3. Calculation using formula.

As the given right-angled triangle has height h and base h+6, so the area of a triangle is

$\begin{array}{l}A=\frac{1}{2}\times h\times \left(h+6\right)\\ A=\frac{1}{2}\left(h^2+6h\right)\\ A=\frac{h^2}{2}+3h\end{array}$

Now, if $h=12\text{\hspace{0.17em}inches}$, the area is

$\begin{array}{l}A=\frac{{\left(12\right)}^2}{2}+3\left(12\right)\\ A=\frac{12\times 12}{2}+36\\ A=6\times 12+36\\ A=72+36\\ A=108{\text{\hspace{0.17em}inches}}^2\end{array}$