Inscribed circle of a triangle, step by step

The circle inside a right triangle touches all three sides at once.

The question

The circle inside a 27-36-45 triangle

from the right angle along each leg=r

(27 − r) + (36 − r)=45

(27 + 36 − 45) ÷ 2=9

The circle touches all three sides, so the two pieces of it nearest the right angle make a little square with the two legs: from the corner along each leg is exactly r, the radius.
That leaves 27 − r on one leg and 36 − r on the other. Those two pieces reach the far corners — and from a corner the two touching lengths are equal, so together they are exactly the hypotenuse: (27 − r) + (36 − r) = 45.
Rearranged that is 27 + 36 − 45 = 2r, so r = (27 + 36 − 45) ÷ 2 = 9. The circle inside has a radius of 9 cm — it reaches 18 cm short of one corner and 27 cm short of the other.

How it works.

01
The circle touches all three sides, so the two pieces of it nearest the right angle make a little square with the two legs: from the corner along each leg is exactly r, the radius.
02
That leaves 27 − r on one leg and 36 − r on the other. Those two pieces reach the far corners — and from a corner the two touching lengths are equal, so together they are exactly the hypotenuse: (27 − r) + (36 − r) = 45.
03
Rearranged that is 27 + 36 − 45 = 2r, so r = (27 + 36 − 45) ÷ 2 = 9. The circle inside has a radius of 9 cm — it reaches 18 cm short of one corner and 27 cm short of the other.