Chapter 10.1, Problem 24PSC

To determine

To Find : Radius of the circle.

Answer to Problem 24PSC

  $\text{Radius of circle is 6}\text{.623}\text{.}$

Explanation of Solution

Given information :

  $\begin{array}{l}\text{An isosceles trianle with each leg measuring 13 is inscribed in a circle}\text{. Altitude to}\\ \text{the of triangle is 5}\text{.}\end{array}$

  

  $\begin{array}{}$ \begin{array}{l}\text{Draw }OA,OB\text{and }OC,\text{O is centre of circle}\text{.}\\ \text{D is point on BC, joining OD}\text{. Let X be the}\\ \text{measure of OD}\text{.}\\ \text{In triangle BAD,}\end{array}$$ {(r+x)}^2+{(2.5)}^2={13}^2$$ \left[\because BD=\frac{BC}{2}=\frac{5}{2}=2.5\right]$$ {(r+x)}^2+6.25=169$$ {(r+x)}^2=169-6.25=162.75$$ (r+x)=\sqrt{162.75}=12.757\mathrm{....}(1)$$ \text{In triangle }BOD\text{,}$$ r^2={(2.5)}^2+x^2$$ \begin{array}{l}{r}^2=6.25+x^2\\ r^2-x^2=6.25\\ (r-x)(r+x)=6.25\mathrm{.....}(2)\\ \text{Using (1) and (2)}\\ (r-x)12.757=6.25\\ r-x=\frac{6.25}{12.757}=0.489\mathrm{....}(3)\\ \text{Solving (1) and (3)}\\ 2r=13.247\\ r=6.623\\ \text{Radius of circle is 6}\text{.623}\end{array}$\end{array}$