In an isosceles trapezoid, the diagonal is perpendicular to the lateral side.

In an isosceles trapezoid, the diagonal is perpendicular to the lateral side. Find the area of a trapezoid if the side is 6 centimeters and one of the corners of the trapezoid is 60 degrees.

1. A, B, C, D – the tops of the trapezoid. AB = 6 cm. BE – height. S – area. ∠А = 60 °. Diagonal BD is perpendicular to AB. ∠ABD = 60 °.

2.∠ADB = 180 ° – 90 ° – 60 ° = 30 °.

3. Cathet AB is against ∠ADB, equal to 30 °. Therefore, its length is equal to 1/2 of the hypotenuse AD.

AD = AB x 2 = 6 x 2 = 12 cm.

4. BE / AB = sine ∠ABD = sine 60 ° = √3 / 2.

BE = AB x √3 / 2 = 6 x √3 / 2 = 3√3 cm.

5. AE = √AB² – BE² = √36 – 27 = √9 = 3 cm.

6. AE = (AD – BC) / 2.

BC = 12 – 6 = 6 cm.

6. S = (AD + BC) / 2 x BE = (12 + 6) / 2 x 3√3 = 27√3 cm².