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