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

In a rectangular trapezoid, the smaller base is 3 cm, the large side is 4 cm, and one of the corners of the trapezoid is 150 degrees, find the area of the trapezoid.

Rectangular is a trapezoid in which one side is perpendicular to its bases.

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, you need to find the length of the larger base AD and the height BH.

Consider the triangle ΔАВН.

To calculate the height of the HV, we will use the sinus torus. The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:

sin A = BH / AB;

BH = AB ∙ sin A.

Since the sum of the angles of a trapezoid adjacent to one side is 180 °, then:

∠А = 180º – ∠В;

∠А = 180º – 150º = 30º;

sin 30º = 1/2;

BH = 4 1/2 = 4/2 = 2 cm.

In order to find the length of the larger base, we find the segment AH. Let’s apply the Pythagorean theorem:

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

AH ^ 2 = AB ^ 2 – BH ^ 2;

AH ^ 2 = 4 ^ 2 – 2 ^ 2 = 16 – 4 = 12;

AH = √12 ≈ 3.5.

In order to find the length of a larger base, you need:

AD = AH + HD;

AD = 3.5 + 3 = 6.5 cm.

S = (3 + 6.5) / 2 2 = 9.5 cm2.

Answer: the area of ​​the trapezoid is 9.5 cm2.