A straight prism is based on a rhombus with an acute angle of 60 degrees and a side of 8 cm. Find the smaller diagonal of the prism if its lateral edge is 6 cm.
Draw the AC and BD diagonals at the base of the prism. Since there is a rhombus at the base of the prism, the diagonals are divided in half at the point of intersection, BO = DO, AO = CO.
The diagonals of the rhombus are also the bisector of the angle, then the angle BAO = BAD / 2 = 60/2 = 30.
The leg BO of the right-angled triangle AOB lies opposite the angle 30, which means it is equal to half the length of the hypotenuse AB. BО = AB / 2 = 8/2 = 4 cm.
Then the diagonal of the rhombus BD = 2 * BO = 2 * 4 = 8 cm.
Consider a right-angled triangle В1ВD, and by the Pythagorean theorem define the diagonal В1D.
B1D ^ 2 = B1B ^ 2 + BD ^ 2 = 6 ^ 2 + 8 ^ 2 = 36 + 64 = 10.
B1D = 10 cm.
Answer: The smaller diagonal of the prism is 10 cm.
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