The bases of the trapezoid are 16 and 18, one of the lateral sides is 4√2

The bases of the trapezoid are 16 and 18, one of the lateral sides is 4√2, the angle between it and one of the base is 135 degrees. Find the area of the trapezoid.

The obtuse angle of a trapezoid is the angle between the lateral side and the smaller base. Consider the triangle formed by this side, the smaller base, and the diagonal opposite this obtuse angle. Its area can be found as half the product of the lengths of the lateral side and the smaller base by the sine of the angle between them:

S = 0.5 * 16 * 4√2 * sin 135 ° = 32√2 * √2 / 2 = 32.

On the other hand, the area of ​​this triangle is equal to half of the product of the smaller base and the height, drawn from the opposite angle to the straight line that is its continuation:

S = 0.5 * h * 16;

h = 2 * S / 16 = 64/16 = 4.

Obviously, the height of this triangle is equal to the height of the trapezoid.

The area of ​​the trapezoid is equal to the product of the half-sum of the bases and the height:

Str = 4 * (16 + 18) / 2 = 68.