# In the rectangular parallelepiped ABCDA1B1C1D1, the ratios of the lengths of the edges and the diagonal are known: AB: BC:

In the rectangular parallelepiped ABCDA1B1C1D1, the ratios of the lengths of the edges and the diagonal are known: AB: BC: CA1 = 9: 12: 17. The sum of the lengths of all the edges of the parallelepiped is 841. Find the length of the edge AA1.

Let the edge length AB = 9 * X cm, then BC = 12 * X cm, CA1 = 17 * X cm.

In a right-angled triangle ABC, we determine the length of the hypotenuse AC.

AC ^ 2 = AB ^ 2 + BC ^ 2 = 81 * X ^ 2 + 144 * X ^ 2 = 225 * X ^ 2.

AC = 15 * X ^ 2.

In a right-angled triangle AA1C, according to the Pythagorean theorem, AA1 ^ 2 = CA1 ^ 2 – AC ^ 2 = 289 * X ^ 2 – 225 * X ^ 2 = 64 * X ^ 2.

AA1 = 8 * X ^ 2.

The sum of the lengths of all edges is: 841 = 4 * (AB + BC + AA1) = 4 * (9 * X + 12 * X + 8 * X) = 116 * X.

X = 841/116 = 29/4.

AA1 = 8 * X = 8 * 29/4 = 58 cm.

Answer: The length of the rib AA1 is 58 m.

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