Chapter 16.6, Problem 5PS

To determine

To find the length of the third side of the triangle shown.

Explanation of Solution

Given:

The given triangle is

  

Calculation:

Finding the length of the side using Stewart’s theorem as follows-

  

Here using the angle bisector theorem-

  $\begin{array}{l}\frac{a}{b}=\frac{m}{n}\\ \frac{3}{5}=\frac{m}{n}\end{array}$

Now, using the Stewart’s theorem-

  $\begin{array}{l}{a}^{2}n+b^{2}m=c{d}^2+cmn\\ a^2+b^2\frac{m}{n}=c\left[\frac{d^2}{n}+m\right]\{\text{dividing by n}}\\ c=\frac{a^2+b^2\frac{m}{n}}{\frac{d^2}{n}+m}=\frac{25+64\times \frac{3}{5}}{\frac{36\times 3}{5m}+m}=\frac{63.4}{\frac{21.6}{m}+m}\\ m+n=\frac{63.4}{\frac{21.6}{m}+m}\\ m+\frac{5m}{3}=\frac{63.4}{\frac{21.6}{m}+m}\\ \left(\frac{21.6}{m}+m\right)\left(m+\frac{5m}{3}\right)=63.4\\ 21.6+36+m^2+\frac{5m^2}{3}=63.4\\ \frac{8m^2}{3}=5.8\\ m^2=2.175\\ m=1.48\end{array}$

Therefore third side $c=\frac{8m}{3}=\frac{8\times 1.48}{3}=3.95$

To determine

To find the circumradius of the triangle to four significant digits.

Explanation of Solution

Given:

The given triangle is-

  

Calculation:

The circumradius is given by the formula is

  $=\frac{abc}{4A}=\frac{5\times 8\times 3.95}{4\times 7.956}=4.965$