In an isosceles trapezoid, the bases are 10 and 4, and the acute angle is 45 degrees. Find the area of the trapezoid.

The area of the trapezoid is equal to the product of the half-sum of its bases by the height:

S = (a + b) / 2 h.

Consider the triangle ΔАВН. Since the trapezoid is isosceles, then:

AH = (AD – BC) / 2;

AH = (10 – 4) / 2 = 6/2 = 3 cm.

To calculate the ВН height, we use the tangent of the angle ∠A, which is the ratio of the opposite leg to the adjacent one:

tg A = BH / AH;

BH = AH · tg A;

tg 45º = 1;

BH = 3 1 = 3 cm.

S = (10 + 4) / 2 3 = 7 3 = 21 cm.

Answer: the area of the trapezoid is 21 cm2.