StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q48E

Expert-verifiedFound in: Page 308

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 $.

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

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

$r=\frac{\frac{3+4+5}{2}}{\frac{3.4}{2}}\phantom{\rule{0ex}{0ex}}r=\frac{6}{6}\phantom{\rule{0ex}{0ex}}r=1,\phantom{\rule{0ex}{0ex}}$

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

${\mathrm{I}}^{2}.\mathrm{\pi}=\mathrm{\pi}$.

94% of StudySmarter users get better grades.

Sign up for free