The bases of the trapezium are 11 and 23, the lateral side is equal to 10, forms an angle of 150 °

The bases of the trapezium are 11 and 23, the lateral side is equal to 10, forms an angle of 150 ° with one of the bases of the trapezoid. Find the area of the trapezoid.

A trapezoid is a quadrilateral in which one pair of opposite sides is parallel, and the sides are not equal to each other.

The area of ​​a trapezoid is the product of the half-sum of its bases by the height:

S = (a + b) / 2 h, where:

S is the area of ​​the trapezoid;

a – smaller base;

b – larger base;

h is the height of the trapezoid.

To do this, find the height of the HV. Consider the triangle ΔАВН. This triangle is rectangular.

Let’s apply the sine theorem. The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:

sin A = BH / AB;

ВН = AB sin A.

Since the sum of the inner angles of the trapezoid adjacent to the lateral side is 180 °:

∠А = 180º – ∠В;

∠А = 180º – 150º = 30º;

sin 30º = 1/2;

BH = 10 1/2 = 5 cm.

S = (11 + 23) / 2 ∙ 5 = 34/2 · 5 = 85 cm2.

Answer: the area of ​​the trapezoid is 85 cm2.