Find all sides of an isosceles trapezoid if its upper base is 60 cm, height 12 cm, and the angle at the base is 60˚

Isosceles is a trapezoid in which the sides are equal:

AB = CD.

Consider a triangle ∆АВН, formed by the height ВН. To calculate the hypotenuse AB, we use the theorem of sines. The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:

sin A = BH / AB;

AB = BH / sin A;

sin 60 ° = 0.866;

AB = 12 / 0.866 = 13.86 cm;

CD = AB = 13.86 cm.

Since the segment of the larger base, located between the heights of the trapezoid, is equal to the length of the smaller base, then:

AD = AH + NK + KD;

AH = KD.

To calculate the length of AH, we apply the cosine theorem. The cosine of an acute angle of a right-angled triangle is the ratio of the adjacent leg to the hypotenuse:

cos A = AH / AB;

AH = AB · cos A;

cos 60 ° = 1/2;

AH = 13.86 1/2 = 6.93 cm.

AD = 60 + 6.93 + 6.93 = 73.86 cm.

Answer: sides AB = CD are 13.86 cm, base AD is 73.86 cm.