Find the area of a right-angled triangle if the radius of the circumscribed circle around it is 5

Find the area of a right-angled triangle if the radius of the circumscribed circle around it is 5 and the radius of the inscribed circle is 2.

Let the triangle ABC be given.
Since the center of the circumscribed circle is the middle of the hypotenuse, it means that the hypotenuse AB = 5 * 2 = 10.

The touch point L of the inscribed circle with the hypotenuse divides the hypotenuse into segments BL = x and AL = 2 * R – x = 10 – x.
The touch point D of the inscribed circle with the leg BC divides the leg into segments BD = x and DC = r = 2.
The touch point R of the inscribed circle with the leg AC divides the leg into segments AK = AL = 10 – x and KC = r = 2.
Let’s find the legs.
On the one hand, BC = x + r = x + 2.
On the other hand, by the Pythagorean theorem:
BC ^ 2 = AB ^ 2 – AC ^ 2 = 100 – (AK + KC) ^ 2 = 100 – (10 – x + 2) ^ 2 = 100 – (12 – x) ^ 2.

(x + 2) ^ 2 = 100 – (12 – x) ^ 2.
X ^ 2 – 10 * x + 24 = 0.
(x – 5) ^ 2 = 1.
X – 5 = ± 1.
X1 = 6.
X2 = 4.

Then one of the legs is x + 2 = 4 + 2 = 6, and the other 10 – x + 2 = 10 – 4 + 2 = 8.
Area of ​​a right triangle
S = ½ * BC * AC = ½ * 6 * 8 = 24.