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

In an isosceles trapezoid, the diagonal is perpendicular to the lateral side. Find the area of the trapezoid if the larger base = 16√3 and one of the corners of the trapezoid = 60 degrees.

1. A, B, C, D – the tops of the trapezoid. АD = 16√3 cm. ∠D = 60 °. The AC diagonal is perpendicular to the CD.

2. AC: AD = 60 ° sine. AC = AD x √3 / 2 = 16√3 x √3 / 2 = 24 cm.

3. We calculate the length of the CH through the sine ∠CAН. ∠CAН = 180 ° – 90 ° – 60 ° = 30 °.

CH: AC = sine 30 ° = 1/2.

CH = 24 x 1/2 = 12 cm.

4. AН: AC = cosine ∠СAН = cosine 30 ° = √3 / 2.

AH = AC x √3 / 2 = 24 x √3 / 2 = 12√3 cm.

5. According to the properties of an isosceles trapezoid, (АD + ВС) / 2 = АН = 12√3 cm.

6. Area of the trapezoid = (BC + AD) / 2 x CH = 12√3 x 12 = 144√3 cm².