# 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 °.

August 31, 2021 | education

| 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.

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