The height CK of the right-angled triangle MCP, lowered to the hypotenuse of the MP, is 7 m

The height CK of the right-angled triangle MCP, lowered to the hypotenuse of the MP, is 7 m, and the projection of the CP leg to the hypotenuse is 9 m. Find all the sides of this triangle.

The height CK of the right-angled triangle MCR divides it into two right-angled triangles MCK and RSK.

The projection of the CP segment onto the MR hypotenuse is the RC = 9 m segment.

From the right-angled triangle PCK, by the Pythagorean theorem, we determine the length of the hypotenuse CP.

CP ^ 2 = RK ^ 2 + CK ^ 2 = 81 + 49 = 130.

СР = √130 m.

The CK height is drawn to the hypotenuse from the vertex of the right angle, then the square of the CK height is equal to the product of the segments by which the height divides the MR hypotenuse.

CK ^ 2 = MK * MR.

MK = CK ^ 2 / MR = 49/9 m.

Then MR = MK + RK = 49/9 + 9 = 130/9 = 14 (4/9) m.

From a right-angled triangle CMK, by the Pythagorean theorem, CM ^ 2 = CK ^ 2 + MK ^ 2 = 49 + 2401/81 = 6370/81.

CM = 7 * √130 / 9 m.

Answer: The lengths of the sides of the triangle are 7 * √130 / 9 m, 14 (4/9) m, √130 m.