Chapter 5.6, Problem 27E

Solution

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

The triangle is possible and the area of the triangle is $314.05{\text{unit}}^2$ .

Given information:

The length of the sides of the triangle are $a=33.4\text{units}$ , $b=28.5\text{units}$ , and $c=22.3\text{units}$ .

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 $33.4$ for $a$ , $28.5$ for $b$ and $22.3$ for $c$ in the three conditions to check whether triangle possible.

  $\begin{array}{l}a+b>c\\ 33.4+28.5>22.3\end{array}$

This condition is satisfied.

  $\begin{array}{l}b+c>a\\ 28.5+22.3>33.4\end{array}$

This condition is satisfied.

  $\begin{array}{l}a+c>b\\ 33.4+22.3>28.5\end{array}$

This condition is satisfied.

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

Substitute $33.4$ for $a$ , $28.5$ for $b$ and $22.3$ for $c$ in the formula $s=\frac{a+b+c}{2}$ , to calculate the semi perimeter.

  $\begin{array}{c}s=\frac{33.4+28.5+22.3}{2}\\ =42.1\end{array}$

Substitute $33.4$ for $a$ , $28.5$ for $b$ and $22.3$ for $c$ , and $42.1$ for $s$ into the Heron’s formula to calculate the area of triangle.

  $\begin{array}{c}\text{Area}=\sqrt{42.1\left(42.1-33.4\right)\left(42.1-28.5\right)\left(42.1-22.3\right)}\\ =\sqrt{98629.18}\\ =314.05\end{array}$

Therefore, the area of the triangle is $314.05{\text{unit}}^2$ .