In an isosceles trapezoid, the smaller base is 4 cm, the lateral side is 6 cm

In an isosceles trapezoid, the smaller base is 4 cm, the lateral side is 6 cm, and one of the corners is 150 degrees. Find the area of the trapezoid.

Let’s denote the apex of the trapezoid by ABCD. Then the lesser base will be BC = 4, AB = CD = 6. ∠ B = ∠ C = 150 °.

The bases of the trapezoid are parallel and the lateral side is secant. D, ∠C – inner one-sided corners. It is known that the sum of the inner one-sided angles is 180 °.

Therefore: ∠ D + ∠ C = 180 ° => ∠ D = 180 ° – 150 ° = 30 °.

The height dropped from the vertex C to the base (to the point H) forms a right-angled triangle, the acute angles of which are 60 and 30 degrees. The height (CH) also lies opposite an angle of 30 ° and is equal to half the length of the hypotenuse (CD), the length of which is 6 cm. Hence, we have that CH = 3 cm.

The height dropped from the tops of the trapezoid forms a rectangle with sides 4 by 3 cm. Segment HD = AM = 6 * cos 30 ° = 6 * √3 / 2 = 3√3. => AD = 4 + 6√3.

S = ((AD + BC) / 2) * CH = ((4 + 4 + 6√3) / 2) * 3 = 12 + 9√3.

Answer. 12 + 9√3 (cm2).

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