Triangle ABC is isosceles with base AC. Find the base if the Median BM is 12 cm and the side is 13 cm.

Since triangle ABC is isosceles, its median BM is also its height. This means that triangle ABM is right-angled with right angle M. AB is the hypotenuse of this triangle, we know its value. We also know the BM leg. Using the Pythagorean theorem, you can find the second leg AM. Let’s calculate it:

AM = √ (AB² – BM²) = √ (13² – 12²) = √ (169 – 144) = √25 = 5 cm.

AM is equal to half the base of triangle ABC, since the median BM bisects it. So, to find AC, you need to multiply AM by 2. Let’s calculate its value:

AC = 2AM = 2 * 5 = 10 cm.

Answer: The base AC of an isosceles triangle ABC is 10 cm.