Chapter 5.6, Problem 22E

Solution

To find: Whether 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 $17.41{\text{unit}}^2$ .

Given information:

The length of the sides of the triangle are $a=5\text{units}$ , $b=9\text{units}$ , and $c=7\text{units}$ .

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 $5$ for $a$ , $9$ for $b$ and $7$ for $c$ in the three conditions to check whether triangle possible.

  $\begin{array}{l}a+b>c\\ 5+9>7\end{array}$

This condition is satisfied.

  $\begin{array}{l}b+c>a\\ 9+7>5\end{array}$

This condition is satisfied.

  $\begin{array}{l}a+c>b\\ 5+7>9\end{array}$

This condition is satisfied.

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

Substitute $5$ for $a$ , $9$ for $b$ and $7$ for $c$ in the formula $s=\frac{a+b+c}{2}$ , to calculate the semi perimeter.

  $\begin{array}{c}s=\frac{a+b+c}{2}\\ =\frac{5+9+7}{2}\\ =10.5\end{array}$

Substitute $5$ for $a$ , $9$ for $b$ and $7$ for $c$ , and $10.5$ 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.

  $\begin{array}{c}\text{Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\\ =\sqrt{10.5\left(10.5-5\right)\left(10.5-9\right)\left(10.5-7\right)}\\ =\sqrt{303.18}\\ =17.41\end{array}$

Therefore, the area of the triangle is $17.41{\text{unit}}^2$ .