Question

What are the lengths of the semi axes of the largest ellipse you can inscribe into a triangle with sides 3,4 , and 5 ? See Exercise 48.

Short answer

Therefore, the length of the semi axes of the largest ellipse is given by,

r =1.

Step-by-step solution

Given

The length of the sides of triangle are 3, 4, and 5.

To find the lengths of the semi axes of the largest ellipse.

We know that the largest ellipse that can be inscribed into any kind of triangle is a circle.

Thus, both of its semi axes are, in fact, its radius.

Same applies particularly for right triangles, as well.

We also know that, for any kind of triangle with side lengths $a,b$ and c, if r is the radius of the circle inscribed into the triangle, and $s=\frac{a+b+c}{2}$, then for the triangle's area P applies ,P=rs which leads to $r=\frac{S}{P}$.

A triangle with side lengths 3,4 , and 5 is a right triangle, so we have

$$\begin{gathered} r=\frac{{\displaystyle \frac{3+4+5}{2}}}{{\displaystyle \frac{3.4}{2}}} \\ r=\frac{6}{6} \\ r=1. \end{gathered}$$