## 71. Math Puzzle $4 / 71$

The side lengths of a triangle are $a=4 \mathrm{~cm}, b=13 \mathrm{~cm}$ and $c=15 \mathrm{~cm}$.

What are a) the area, b) the three heights, c) the radius of the incircle, and d) the radii of the three excircles?
The perimeter is $u=32 \mathrm{~cm}$, the semi-perimeter $s=\frac{u}{2}=16 \mathrm{~cm}$. According to Heron's formula for the area of a triangle, $A=\sqrt{s(s-a)(s-b)(s-c)}=24 \mathrm{~cm}^{2}$.

The area can also be expressed as $A=\frac{1}{2} h g$. From this, the heights are $h_{a}=12 \mathrm{~cm}, h_{b}=$ $3.7 \mathrm{~cm}$ and $h_{c}=3.2 \mathrm{~cm}$.

From the formula for the area of a triangle $A=\rho \cdot s$, the radius $\rho$ of the incircle is: $\rho=$ $1.5 \mathrm{~cm}$. The radii of the excircles are determined from $A=\rho_{a}(s-a)=\rho_{b}(s-b)=\rho_{c}(s-c)$ to be $\rho_{a}$ $=2 \mathrm{~cm}, \rho_{b}=8 \mathrm{~cm}$ and $\rho_{c}=24 \mathrm{~cm}$.