Chapter 5.6, Problem 24E

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 $113.84$ square units.

Given information:

The length of the sides of the triangle are $a=23\text{units}$ , $b=19\text{units}$ , and $c=12\text{units}$ .

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 $23$ for $a$ , $19$ for $b$ and $12$ for $c$ in the three conditions to check whether triangle possible.

  $\begin{array}{l}a+b>c\\ 23+19>12\end{array}$

This condition is satisfied.

  $\begin{array}{l}b+c>a\\ 19+12>23\end{array}$

This condition is satisfied.

  $\begin{array}{l}a+c>b\\ 23+12>19\end{array}$

This condition is satisfied.

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

Substitute $23$ for $a$ , $19$ for $b$ and $12$ for $c$ in the semi-perimeter formula, to calculate the semi perimeter.

  $\begin{array}{c}s=\frac{23+19+12}{2}\\ =27\end{array}$

Substitute $23$ for $a$ , $19$ for $b$ and $12$ for $c$ , and $27$ for $s$ into the Heron’s formula to calculate the area of triangle.

  $\begin{array}{c}\text{Area}=\sqrt{27\left(27-23\right)\left(27-19\right)\left(27-12\right)}\\ =\sqrt{12960}\\ =113.84\end{array}$

Therefore, the area of the triangle is $113.84{\text{unit}}^2$ .