Chapter 8.4, Problem 53AYU

To determine

To prove: The formula $K=rs$ and $r=\sqrt{\frac{\left(s-a\right)\left(s-b\right)\left(s-c\right)}{s}}$ .

Answer to Problem 53AYU

The formula $K=rs$ and $r=\sqrt{\frac{\left(s-a\right)\left(s-b\right)\left(s-c\right)}{s}}$ are proved.

Explanation of Solution

Given:

It is given that $s=\frac{1}{2}\left(a+b+c\right)$ .

Calculation:

Express the area $K$ of $\Delta PQR$ as a sum of the three areas as follows.

  $\begin{array}{c}K=K_{\Delta OPQ}+K_{\Delta OQR}+K_{\Delta ORP}\\ =\frac{PQ\cdot r}{2}+\frac{RQ\cdot r}{2}+\frac{PR\cdot r}{2}\\ =\frac{cr}{2}+\frac{ar}{2}+\frac{br}{2}\\ =\frac{r\left(a+b+c\right)}{2}\end{array}$

Substitute $s$ for $\frac{1}{2}\left(a+b+c\right)$ in the above equation.

  $K=rs$

To prove the second formula, use Heron’s formula for $K$ and replace $K$ by $rs$ as shown below.

  $\begin{array}{c}K=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\\ rs=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\\ r=\frac{\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}{s}\\ r=\sqrt{\frac{\sqrt{\left(s-a\right)\left(s-b\right)\left(s-c\right)}}{s}}\end{array}$

Hence, the formula $K=rs$ and $r=\sqrt{\frac{\left(s-a\right)\left(s-b\right)\left(s-c\right)}{s}}$ are proved.