The circle passes through vertex A of rectangle ABCD and touches sides BC and CD at points

The circle passes through vertex A of rectangle ABCD and touches sides BC and CD at points M and N, respectively. It is known that BM = 24, DN = 7. Find the area of the rectangle ABCD.

BC – tangent to the circle. The tangent and the radius of the circle drawn to the tangent point are perpendicular. ON ┴BC. DC is also tangent to the circle, and OF┴DC.

Extend the radii ON and OF to the intersection with the sides of the rectangle. The point of intersection with the side AB will be denoted by K, and with the side AD will be denoted by M.

Find the radius of the circle OA by the Pythagorean theorem from the triangle AOK.

AO ^ 2 = AK ^ 2 + OK ^ 2; AK = DF = 7; OK = BN = 24;

AO ^ 2 = 7 ^ 2 + 24 ^ 2 = 49 + 576 = 625; AO = √625 = 25;

ON = OF = AO = 25 are the radii of the circle.

Find the sides of the rectangle AB = MN = MO + ON; MO = DF = 7; AB = 7 + 25 = 32.

BC = OK + OF; BC = 24 + 25 = 49.

The area of ​​a rectangle is equal to the product of its sides.

S = AB * BC;

S = 32 * 49 = 1568,

Answer. 1568.