The segment BM is perpendicular to the plane of the rectangle ABCD. Find the MD if AB = 3 cm

The segment BM is perpendicular to the plane of the rectangle ABCD. Find the MD if AB = 3 cm, BC = √7 cm, and MB = 3 cm.

The opposite sides of the rectangle are equal, which means AB = CD = 3 cm, BC = AD = √7 cm.

Two adjacent sides and the diagonal of the rectangle form a right-angled triangle in which the diagonal is the hypotenuse. Therefore, by the Pythagorean theorem:

AB ^ 2 + AD ^ 2 = BD ^ 2;

BD ^ 2 = 3 ^ 2 + (√7) ^ 2 = 9 + 7 = 16 = 42;

BD = 4 cm.

Since BM is perpendicular to the plane of the rectangle, it means that the BMD triangle is rectangular with the hypotenuse MD. Hence:

MD ^ 2 = BM ^ 2 + BD ^ 2 = 3 ^ 2 + 4 ^ 2 = 9 + 16 = 25 = 52;

MD = 5 cm.