The base of the trapezoid is 31 and 87, one of the sides is 45 and the cosine of the angle between

The base of the trapezoid is 31 and 87, one of the sides is 45 and the cosine of the angle between it and one of the base is 0.6. find the area of the trapezoid

1. Let’s draw the height BH. Consider △ AHB: ∠AHB = 90 ° (since BH is the height), AB = 45 is the hypotenuse (since it lies opposite the right angle), AH and BH are legs, cos∠BAH = 0.6.

In a right-angled triangle, the cosine of an acute angle is equal to the ratio of the leg adjacent to the given angle to the hypotenuse.

Thus, cos∠BAH = AH / AB.

Substitute the data for the value condition and find the length AH:

AH / 45 = 6/10;

AH = (45 * 6) / 10 = 27.

2. By the Pythagorean theorem, we find the length BH:

BH = √ (AB² – AH²) = √ (45² – 27²) = √ (2025 – 729) = √1296 = 36.

3. Find the area of ​​the trapezoid ABCD by the formula:

S = (a + b) / 2 * h,

where a and b are the lengths of the bases, h is the length of the height.

Thus:

S = (87 + 31) / 2 * 36 = 118/2 * 36 = 59 * 36 = 2124.

Answer: S = 2124.