In a rectangular trapezoid, the smaller base is 3cm, the larger side is 4cm

In a rectangular trapezoid, the smaller base is 3cm, the larger side is 4cm, and one of the corners of the trapezoid is 150 degrees. 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 ​​the trapezoid is equal to 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, b – the base of the trapezoid;

h – height.

If the perpendiculars of the trapezoid are lowered from obtuse angles to the larger base, then the segment located between them is equal to the length of the smaller base:

HD = BC.

Consider the triangle ΔАВН. This triangle is right-angled with a right angle ∠AНВ.

Let’s calculate the degree measure of the angle ∠АВН. Since the angle ∠НВС = 90º, then:

∠AВН = 150º – 90º = 60º.

Using the cosine theorem, we find the length of the BH height.

The cosine of an acute angle of a right-angled triangle is the ratio of the adjacent leg to the hypotenuse:

cos B = BH / AB;

cos 60º = 1/2;

BH = AB · cos B;

BH = 4 ∙ 1/2 = 4/2 = 2 cm.

In order to find the length of the larger base, we calculate the segment AH. To do this, we apply the Pythagorean theorem, according to which the square of the hypotenuse is equal to the sum of the squares of the legs:

AB ^ 2 = BH ^ 2 + AH ^ 2;

AH ^ 2 = AB ^ 2 – BH ^ 2;

AH ^ 2 = 4 ^ 2 + 2 ^ 2 = 16 – 4 = 12;

AH = √12 = 3.46 cm.

AD = AH + HD;

AD = 3.46 + 3 = 6.46 cm.

S = (3 + 6.46) / 2 2 = 9.46 cm2.

Answer: the area of ​​the trapezoid is 9.46 cm2.