# In the AMPK parallelogram, the diagonal MK is perpendicular to the side AM

In the AMPK parallelogram, the diagonal MK is perpendicular to the side AM, the diagonals intersect at point O. Find the side MA if the diagonals MP and AP are 18 and 82, respectively.

1. In the condition of the problem in the fourth line from the top, the diagonal МР is mistakenly indicated, you need MK.

2. Applying the parallelogram property that the diagonals at the intersection point are halved, we calculate the length of the segments AO and MO:

MO = 18: 2 = 9 cm.

AO = 82: 2 = 41 cm.

3. In a right-angled triangle AMO, the sides MO and AO are legs, and the AO side is the hypotenuse.

4. Applying the Pythagorean theorem, we calculate the length of AM:

AM = √AO ^ 2 – MO ^ 2 = √41 ^ 2 – 9 ^ 2 = √1681 – 81 = √1600 = 40 cm.

Answer: AM length is 40 cm.

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