Question

What is the length of the diameter of a circle inscribed in a right triangle with the length of hypotenuse c and the sum of the legs' lengths equal to m?

Answer 1

Answer:

  diameter = m - c

Step-by-step explanation:

In ΔABC, let ∠C be the right angle. The length of the tangents from point C to the inscribed circle are "r", the radius. Then the lengths of tangents from point A are (b-r), and those from point B have length (a-r).

The sum of the lengths of the tangents from points A and B on side "c" is ...

  (b-r) +(a-r) = c

  (a+b) -2r = c

Now, the problem statement defines the sum of side lengths as ...

  a+b = m

and, of course, the diameter (d) is 2r, so we can rewrite the above equation as ...

  m -d = c

  m - c = d . . . . add d-c

The diameter of the inscribed circle is the difference between the sum of leg lengths and the hypotenuse.