Find the radius of a circle inscribed in a right-angled triangle if the radii of the circles inscribed

Find the radius of a circle inscribed in a right-angled triangle if the radii of the circles inscribed in the triangles into which it is divided by the height drawn to the hypotenuse are 4 and 5.

Since the height of the CH of a right-angled triangle is drawn from the apex of the right angle, the formed triangles ACN and BCH are similar to each other and are similar to the ABC triangle in the first sign.

Then, in such triangles, the radii of the inscribed circles are proportional, then:

AB / R = AC / R1 = BC / R ^ 2.

AC / R1 = BC / R2, then:

AC / 5 = BC / 4.

BC = 4 * AC / 5.

We use the Pythagorean theorem in the ABC triangle.

AB ^ 2 = AC ^ 2 + BC ^ 2 = AC ^ 2 + (4 * AC / 5) ^ 2 = AC ^ 2 + 16 * AC ^ 2/25 = 41 * AC ^ 2/25.

AB = (AC * √41) / 5.

AB / R = AC / R1.

((AC * √41) / 5) / R = AC / R1 = AC / 5.

Let us reduce AC, then:

(√41 / 5) / R = 1/5.

R = √41 cm.

Answer: The radius of the inscribed circle is √41 cm.