Through vertex A of rectangle ABCD, line AK is provided perpendicular to the plane of the rectangle.

Through vertex A of rectangle ABCD, line AK is provided perpendicular to the plane of the rectangle. It is known that KD = 6 cm KB = 7cm KC = 9 cm. Find the distance from K to the plane ABCD and the distance between AK and CD.

Consider the triangle KDC. The segment AD is the projection of the segment KD on the plane of the rectangle ABCD. Then, according to the three perpendicular theorem, KD is perpendicular to CD, which means that the triangle KDC is rectangular. By the Pythagorean theorem, CD ^ 2 = CK ^ 2 – KD ^ 2 = 9 ^ 2 – 6 ^ 2 = 81 – 36 = 45.

СD = 3 * √5 cm.

Similarly, the KBC triangle is rectangular with right angles at the vertex B, then

VS ^ 2 = KS ^ 2 – VK ^ 2 = 9 ^ 2 – 7 ^ 2 = 81 – 49 = 32.

In a right-angled triangle KAD, according to the Pythagorean theorem, we determine the length of the leg AK.

AK ^ 2 = KD ^ 2 – AD ^ 2 = 36 – 32 = 4.

AK = 2 cm.

Answer The length of the segment AK is 2 cm, the length of the side CD is 3 * √5 cm.