In rectangle ABCD, it is known that AB = 2BC. Line FB is perpendicular to the plane

In rectangle ABCD, it is known that AB = 2BC. Line FB is perpendicular to the plane of the rectangle, FB = 7cm, FD = 12cm. Find the sides of a rectangle,

Construct the diagonal BD of the rectangle ABCD.

Triangle BDF is rectangular, in which, according to the Pythagorean theorem, we determine the length of the leg BD.

BD ^ 2 = DF ^ 2 – BF ^ 2 = 144 – 49 = 95.

Triangle ABD is rectangular, in which, by condition, AD = 2 * AB.

Let AD = BC = X cm, then AB = 2 * X cm.

By the Pythagorean theorem, X ^ 2 + 4 * X ^ 2 = 95.

5 * X ^ 2 = 95.

X ^ 2 = 95/5 = 19.

X = AD = √19 cm.

AB = 2 * √19 cm.

Answer: The sides of the rectangle are √19 cm, 2 * √19 cm.