Chapter 5.6, Problem 21E

Solution

To find: If a triangle can be formed using the given lengths and to determine the area of triangle using the Heron’s formula.

The triangle is possible and the area of the triangle is $8.18{\text{unit}}^2$ .

Given information:

The length of the sides of the triangle are $a=4\text{units}$ , $b=5\text{units}$ , and $c=8\text{units}$ .

Formula used:

The conditions to check if the triangle is possible or not are as follows.

  $\begin{array}{l}a+b>c\\ b+c>a\\ a+c>b\end{array}$

The Heron’s formula to determine the area of triangle is

  $\text{Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

Here, $s$ is the semi-perimeter and $s=\frac{a+b+c}{2}$ .

Calculation:

Substitute $4$ for $a$ , $5$ for $b$ and $8$ for $c$ in the three conditions to check whether triangle possible.

  $\begin{array}{c}a+b>c\\ 4+5>8\\ 9>8\end{array}$

This condition is satisfied.

  $\begin{array}{c}b+c>a\\ 5+8>4\\ 13>4\end{array}$

This condition is satisfied.

  $\begin{array}{c}a+c>b\\ 4+8>5\\ 12>5\end{array}$

This condition is satisfied.

Since all the three conditions are satisfied, so the triangle is possible.

Substitute $4$ for $a$ , $5$ for $b$ and $8$ for $c$ in the formula $s=\frac{a+b+c}{2}$ , to calculate the semi perimeter.

  $\begin{array}{c}s=\frac{a+b+c}{2}\\ =\frac{4+5+8}{2}\\ =8.5\end{array}$

Substitute $4$ for $a$ , $5$ for $b$ , $8$ for $c$ , and $8.5$ for $s$ into the formula $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$ , to calculate the area of triangle.

  $\begin{array}{c}\text{Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\\ =\sqrt{8.5\left(8.5-4\right)\left(8.5-5\right)\left(8.5-8\right)}\\ =\sqrt{66.9375}\\ \approx 8.18\end{array}$

Therefore, the area of the triangle is $8.18{\text{unit}}^2$ .