In an isosceles triangle ABC, angle B = 120 degrees, median AM = √21. Find the length of the base of the speaker.

Let the length of the segment BM = CM = X cm, then the length of the segment AB = 2 * X cm.

In triangle ABM we apply the cosine theorem.

AM ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Cos120.

21 = 4 * X ^ 2 + X ^ 2 – 2 * 2 * X * X * (-1/2).

21 = 5 * X ^ 2 + 2 * X ^ 2.

7 * X ^ 2 = 21

X = BM = √3 cm, then AB = 2 * √3 cm.

Let’s build the height BD. Triangle ABD is rectangular with angle A = 30. Cos30 = AD / AB. AD = AB * Cos30 = 2 * √3 * √3 / 2 = 3 cm.

Then AC = 2 * AD = 2 * 3 = 6 cm.

Answer: The length of the base of the speaker is 6 cm.