In an isosceles triangle, the base is equal to the square root of 84, and the angle

In an isosceles triangle, the base is equal to the square root of 84, and the angle at the base is 30. Find the length of the median to the lateral side?

Let’s build the height BH to the base of the AC, which is also the median of the triangle, then AH = AC / 2.

AC = √84 = 2 * √21 cm, then AH = 2 * √21 / 2 = √21 cm.

In a right-angled triangle ABH, BH = AH * tg30 = √21 * √3 / 3 = √3 * √7 * √3 / 3 = √7 cm.

Point O divides the median BH in the ratio of 2/1, then OH = BH / 3 = √7 / 3.

In a right-angled triangle AOH, by the Pythagorean theorem, AO ^ 2 = AH ^ 2 + OH ^ 2 = 21 + 7/9 = 196/9.

AO = 14/3 cm.

ОМ = AO / 2 = 7/3 cm.

Then AM = AO + OM = 14/3 + 7/3 = 21/3 = 7 cm.

Answer: The median length is 7 cm.