In a regular 4-sided prism, the side of the base is 6 cm, the side edge is 8 cm.

In a regular 4-sided prism, the side of the base is 6 cm, the side edge is 8 cm. Find the distance from the side of the base to the diagonal of the prism that does not cross it.

Let us define the length of the hypotenuse DA1 from the right-angled triangle AA1D by the Pythagorean theorem.

DA1 ^ 2 = AA1 ^ 2 + AD ^ 2 = 64 + 36 = 100.

DA1 = 10 cm.

Let’s draw the required height АН and express it from two right-angled triangles, АА1Н and АDН.

Let the length A1H = X cm, then DH = (10 – X) cm.

AH ^ 2 = AA1 ^ 2 – X ^ 2.

AH ^ 2 = AD ^ 2 – (10 – X) ^ 2.

64 – X ^ 2 = 36 – 100 + 20 * X – X ^ 2.

20 * X = 100 + 64 – 36 = 128.

X = A1H = 128/20 = 6.4 cm.

Then AH2 = 64 – 40.96 = 23.04.

AH = 4.8 cm.

Answer: The distance is 4.8 cm.