# Through the middle O of the hypotenuse AB of the right-angled triangle ABC

Through the middle O of the hypotenuse AB of the right-angled triangle ABC, a straight line is drawn, perpendicular to the hypotenuse and intersecting the leg of the AC at point M. Find the area of the triangle AMO if AM = 25 and MC = 7.

Let’s draw a segment BM and consider the triangle AMI, in which OM is the height, since the perpendicular to AB, the median, since OA = OB, therefore, the triangle AMB is isosceles and AM = BM = 25 cm.

Consider a right-angled triangle BCM, in which the angle C is straight, MB = 25 cm, CM = 7 cm. Let us find the leg CB by the Pythagorean theorem. CB ^ 2 = MB ^ 2 – CM ^ 2 = 625 – 49 = 576. CB = 24 cm.

Consider the triangle ABC and find, according to the Pythagorean theorem, the hypotenuse AB, taking into account the fact that the leg AC = AM + CM = 25 + 7 = 32 cm.

AB ^ 2 = AC ^ 2 + BC ^ 2 = 1024 + 576 = 1600. AB = 40 cm.

Then AO = 40/2 = 20 cm.

Consider a right-angled triangle AMO and define the leg MO by the Pythagorean theorem.

MO ^ 2 = AM ^ 2 – AO ^ 2 = 25 ^ 2 – 20 ^ 2 = 225. MO = 15 cm.

Find the area of ​​the AMO triangle.

Samo = (MO * AO) / 2 = 15 * 20/2 = 150 cm2. 