A circle is inscribed in a right-angled triangle with legs of length 5 and 12. What is the area of the quadrilateral formed by the two legs and the two radii to the tangent points on the legs?
4 square units
Hypotenuse \(= \sqrt{5^2 + 12^2} = 13\).
Inradius of a right triangle \(r = \frac{a + b - c}{2} = \frac{5 + 12 - 13}{2} = 2\).
The radii to the tangent points on the legs are perpendicular to the legs, and the legs meet at 90°, so the quadrilateral (right-angle vertex, two tangent points, centre) has all angles 90° and two adjacent sides equal to \(r\); it is a square of side 2.
Area \(= r^2 = 2^2 = 4\) square units.
Hence, the area of the quadrilateral is 4 square units.
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