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?

Question

Please find the diameter using variables from the question above.

Answer 1

I will start you off. Make a drawing. The circle fits inside the right triangle. You can make the legs x and m - x so that they have a sum of m. Draw a diameter that goes from the right angle vertex to the hypotenuse. The diameter will divide the hypotenuse into two parts. You could make those c - y and c. Now you should have two smaller right triangles formed by the diameter. Call that d. All 3 triangles are similar so you can make the corresponding sides proportional. You can also use the Pythagorean Theorem on any of the 3 right triangles.