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Expert-verified Found in: Page 308 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # What is the area of the largest ellipse you can inscribe into a triangle with side lengths 3,4 , and 5 ? Hint: The largest ellipse you can inscribe into an equilateral triangle is a circle.

Therefore, the area of the circle inscribed into this triangle is

${1}^{2}.\pi =\pi$.

See the step by step solution

## Step 1: Given

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

## Step 2: What is the area of the largest ellipse you can inscribe into a triangle.

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

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 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

So, the area of the circle inscribed into this triangle is

${\mathrm{I}}^{2}.\mathrm{\pi }=\mathrm{\pi }$. ### Want to see more solutions like these? 