At the base of a straight prism lies a rhombus with a side of 6 cm and an angle of 60 °. The smaller diagonal

| March 30, 2021 | education

At the base of a straight prism lies a rhombus with a side of 6 cm and an angle of 60 °. The smaller diagonal of the prism is inclined to the base at an angle of 45 °. Find the length of the larger diagonal

Since there is a rhombus at the base of the prism, AB = BC = CD = AD = 6 cm.

In triangle ABD AB = AD, then triangle ABD is isosceles. Since in an isosceles triangle ABD one of the angles is 60, then triangle ABD is equilateral, then BD = 6 cm.

In a right-angled triangle BDD1, the angle DBD1 = 45, then the angle BD1D = 45, and the triangle BDD1 is right-angled and isosceles, then DD1 = BD = 6 cm.

Let us determine the length of the AO segment at the base of the prism. AO is the height of an equilateral triangle, then AO = AB * √3 / 2 = 6 * √3 / 2 = 3 * √3 cm, and the diagonal AC = 2 * AO = 2 * 3 * √3 = 6 * √3 cm.

In a right-angled triangle AA1C, according to the Pythagorean theorem, A1C ^ 2 = AA1 ^ 2 + AC ^ 2 = 6 ^ 2 + (6 * √3) ^ 2 = 36 + 108 = 144.

A1C = 12 cm.

Answer: The length of the larger diagonal of the prism is 12 cm.



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