Two adjacent sides of a parallelogram are respectively equal to two adjacent sides of a rectangle. What is the acute angle of a parallelogram if its area is half the area of the rectangle?
Although both in the rectangle and in the parallelogram the opposite sides are equal, the area of the rectangle is equal to the product of the length and width, and in the parallelogram this must be multiplied by the sine of the angle between them:
Sпр = a • b;
Spair = a • b • sin α.
Since the area of the parallelogram is half the area of the rectangle, then:
Spar = 1/2 Spr;
a • b • sin α = 1/2 • a • b;
sin α = 1/2.
Since 1/2 is equal to a sine of 60 °, the acute angle of the parallelogram is 60 °.
Answer: the angle ∠A is 60 °.
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