A circle inscribed in a right-angled triangle divides the hypotenuse into segments of length 3 and 10.

A circle inscribed in a right-angled triangle divides the hypotenuse into segments of length 3 and 10. Find the area of the triangle.

The radii ОF, OG, OE drawn to the tangency points F, G, E are perpendicular to the tangents (sides) AC, BC, AB.

ОF = OG = OE = r (r – radius);

CF = 3, AF = 10.

ΔABF = ΔAOG (rectangular, common hypotenuse, OF = OG).

AF = AG = 10.

ΔCFO = ΔCEO (rectangular, common hypotenuse, OF = OE).

CF = CE = 3.

Quadrangle BEOG – square (<BEO = <BGO = <B = 90 °, OE = OG).

OG = OE = BE = BG = r;

CB = CE + BE = 3 + r;

AB = AG + GB = 10 + r;

AC = CF + AF = 3 + 10 = 13;

AC ^ 2 = CB ^ 2 + AB ^ 2;

13 ^ 2 = (3 + r) ^ 2 + (10 + r) ^ 2;

169 = 9 + 6r + r ^ 2 + 100 + 20r + r ^ 2;

2r ^ 2 + 26r – 60 = 0;

r ^ 2 + 13r – 30 = 0;

r = (-13 + √ (169 + 120)) / 2 = (-13 + 17) / 2 = 2.

CB = 3 + 2 = 5;

AB = 10 + 2 = 12;

S = (AB * CB) / 2 = (12 * 5) / 2 = 30.

Answer: 30.