Find the area of an isosceles trapezoid if the smaller base is 12 cm and the side is 6 cm and the base angle is 45 °.

Given:
АBСD – isosceles trapezoid,
BH and СK heights,
AB = CD = 6 cm,
BАН = СDК = 45 °
BC = 12 cm
Find S ABCD -?
Solution:
1) triangles ABH = CDK (by angle and hypotenuse). Then AH = KD;
2) Consider a right-angled triangle ABН. We know that the sum of the degree measures of any triangle is 180 degrees. Therefore: angle BAН = angle ABН = 45 °. Then this triangle is also isosceles and AH = HB. Let AH = HB. By the Pythagorean theorem
x ^ 2 + x ^ 2 = 6 ^ 2;
x ^ 2 + x ^ 2 = 36;
2 * x ^ 2 = 36;
x ^ 2 = 36: 2;
x ^ 2 = 18;
x = 3√3 = AH = HB cm;
3) AD = AH + НK + KD;
AD = 3√3 + 12 + 3√3 = 12 + 6√3;
4) S = (12 + 12 + 6√3) / 2 * 3√3 = 9√3 (4 + √3) = 36 √3 + 9 * 3 = 36√3 + 27 square centimeters.
Answer: S АBСD = 36√3 + 27 square centimeters.