We have a triangle $\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$?
I recall that a median is a line segment that joins a vertex to the midpoint of the opposite side. I also recall that the length of a median can be found using Apollonius' theorem, which states that if $AM$ is the median, then $4AM^2 = 2AB^2 + 2AC^2 - BC^2.$ Since $AB = AC = 8$ and $BC = 10,$ I can plug these values into the formula and get $4AM^2 = 2(8^2) + 2(8^2) - 10^2.$ Simplifying, I get $4AM^2 = 128 + 128 - 100 = 156.$ Dividing both sides by 4, I get $AM^2 = 39.$ Taking the square root of both sides, I get $AM = \sqrt{39}.$ 