At the base of the straight prism ABCDA1B1C1D1 lies an isosceles trapezoid, BC || AD, with AB = 3 cm, AD = 5 cm

| June 14, 2021 | education

At the base of the straight prism ABCDA1B1C1D1 lies an isosceles trapezoid, BC || AD, with AB = 3 cm, AD = 5 cm. The diagonal of the prism B1D makes an angle of 45 with the base plane, and the planes AA1B1 and B1BD are perpendicular. Find the volumes of the prism?

Let’s draw a diagonal BD at the base of the prism. Triangle ABD is rectangular, since, by condition, the planes AA1B1 and B1BD are perpendicular, then BD ^ 2 = AD ^ 2 – AB ^ 2 = 25 – 9 = 16.

BD = 4 cm. Let us draw the height of the ВН trapezoid ABCD. Triangles ABD and ABН are similar in acute angle, then AD / AB = BD / BН, BH = AB * BD / AD = 3 * 4/5 = 12/5.

Then AH ^ 2 = AB ^ 2 – BH ^ 2 = 9 – 144/25 = 81/25. AH = 9/5.

Since the trapezoid is isosceles, then BC = AD – 2 * AH = 5 – 2 * 9/5 = 7/5.

Determine the area of ​​the base of the prism.

Sb = (ВС + АD) * ВН / 2 = ((7/5 + 5) * 12/5) / 2 = 192/25 cm2.

In a right-angled triangle В1ВD, the angle В1DВ = 45, then the triangle is isosceles and BB1 = ВD = 4 cm.

Let’s define the volume of the prism.

V = Sbn * BB1 = (192/25) * 4 = 768/25 = 30.72 cm3.

Answer: The volume of the prism is 30.72 cm3.



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