Chapter 11.8, Problem 8PSB

To determine

To find: Measures of all the three altitudes of a triangle with sides 13, 14 and 15.

Answer to Problem 8PSB

The three altitudes of the triangle are 12.92, 12 and 11.2.

Explanation of Solution

Given Information:

Sides of the triangle are:

  $a=13,b=14\text{and}c=15$

Formula used:

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

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

  $a,b\text{and}c$ are the sides of a triangle

Area of a triangle $=\frac{1}{2}\times b\text{'}\times h$

  $b\text{'}$ is the base of the triangle

  $h$ is the altitude of the triangle

Calculation:

The sides of a triangle are given as:

  $a=13,b=14\text{and}c=15$

Using Heron’s formula, Area of a triangle $\left(A_1\right)=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

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

  $\begin{array}{l}s=\frac{13+14+15}{2}\\ s=21\end{array}$

  $\begin{array}{l}\therefore A_1=\sqrt{21\left(21-13\right)\left(21-14\right)\left(21-15\right)}\\ A_1=\sqrt{21\left(8\right)\left(7\right)\left(6\right)}\\ A_1=\sqrt{3\times 7\times 2^3\times 7\times 2\times 3}\\ A_1=\sqrt{3^2\times 7^2\times 2^4}\\ A_1=84\end{array}$

Using the formula for area of triangle,

Area of a triangle $\left(A_1\right)=\frac{1}{2}\times b\text{'}\times h$

  $b\text{'}$ is the base of the triangle

  $h$ is the altitude of the triangle

When base $\left(b\text{'}\right)=13$ and altitude is $h_1$

  $\begin{array}{l}{A}_1=\frac{1}{2}\times b\text{'}\times h_1\\ 84=\frac{1}{2}\times 13\times h_1\\ h_1=\frac{84\times 2}{13}\\ h_1=12.92\mathrm{...}(\text{i})\end{array}$

When base $\left(b\text{'}\right)=14$ and altitude is $h_2$

  $\begin{array}{l}{A}_1=\frac{1}{2}\times 14\times h_2\\ 84=\frac{1}{2}\times 14\times h_2\\ h_2=\frac{84\times 2}{14}\\ h_2=12\mathrm{...}(\text{ii})\end{array}$

When base $\left(b\text{'}\right)=15$ and altitude is $h_3$

  $\begin{array}{l}{A}_1=\frac{1}{2}\times b\text{'}\times h_3\\ 84=\frac{1}{2}\times 15\times h_3\\ h_3=\frac{84\times 2}{15}\\ h_3=11.2\mathrm{...}(\text{iii})\end{array}$

Hence, the three altitudes of the triangle are 12.92, 12 and 11.2