One of the angles of an isosceles trapezoid is 120 degrees. The lengths of the bases of the trapezoid are 7 and 13 cm.

One of the angles of an isosceles trapezoid is 120 degrees. The lengths of the bases of the trapezoid are 7 and 13 cm. What is the perimeter of the trapezoid?

Let ABCD be an isosceles trapezoid, AB = CD, BC = 7 cm – a smaller base, AD = 13 cm – a larger base, angle B = 120 degrees.
1. The sum of all interior angles of a quadrilateral is 360 degrees. Then:
angle A + angle B + angle C + angle D = 360 degrees.
Angle A = angle D = x, angle B = angle C = 120 degrees.
x + 120 + 120 + x = 360;
2x = 360 – 240;
2x = 120;
x = 120/2;
x = 60.
Angle A = angle D = x = 60 degrees.
2. Draw from vertices В and С heights ВН and SK to the base AD. ВН = SK, ВС is parallel to НК, therefore, НВСК is a rectangle (ВС = НК = 7 cm).
Let us find the length of the segment AH.
AD = AH + HK + KD (AH = KD = x);
x + 7 + x = 13;
2x = 13 – 7;
2x = 6;
x = 6/2;
x = 3.
AH = x = 3 cm.
3. Consider a triangle ABN: angle BHA = 90 degrees (since BH is height), angle HAB (angle A) = 60 degrees, AB – hypotenuse (since it lies opposite an angle of 90 degrees), AH = 3 cm – leg.
The cosine of an angle is the ratio of the included angle to the hypotenuse. Then:
cosHAB = AH / AB;
cos60 = 3 / AB;
1/2 = 3 / AB;
AB = 2 * 3;
AB = 6 cm.
AB = CD = 6 cm.
4. The perimeter of the trapezoid ABCD is:
P = AB + BC + CD + AD;
P = 6 + 7 + 6 + 13 = 32 (cm).
Answer: P = 32 cm.