Find the base AD of an isosceles trapezoid ABCD if BC = 10cm, AB = 12cm, angle D = 60 degrees

An isosceles trapezoid is called, in which the sides are equal and the angles at the bases are equal:

AB = CD = 12 cm;

∠A = ∠D = 60º.

Let’s draw the height of the ВН. Since the segment of the base, located between the heights of the trapezoid, is equal to the length of the smaller base, then:

AD = AH + НK + KD;

AH = KD.

Consider the triangle ΔАВН. To calculate AН, we use the cosine theorem, according to which, 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 60º = 1/2;

AH = 12 1/2 = 12/2 = 6 cm.

AD = 6 + 10 + 6 = 22 cm.

Answer: the length of the AD base is 22 cm.