In an isosceles trapezoid, the acute angle is 45 degrees, and the height is 3 times less than the larger base. Find the area of the trapezoid if the smaller base is 6.
We know that the trapezoid is isosceles, and its acute angle is 45 degrees. Therefore, triangles ABE and DFC will be equilateral. Hence, the value BE = AE, and DF = FC. The segments AE = FC are the heights of the trapezoid. We can calculate their value because we know that by condition the height of the trapezoid is 3 times less than the base:
AE = FC = BE = DF = 6: 3 = 2
From here we can calculate the length of the bottom base:
BC = BE + AD + FC = 2 + 6 + 2 = 10.
The area of a trapezoid is equal to the product of the sum of its bases divided by 2 or
S = (BC + AD) * AE / 2 = ((10 +6) * 2) / 2 = 16
Answer: the area of the trapezoid is 16.
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