The perimeter of an isosceles triangle is 36 cm, and its lateral side is 13 cm. Find the median of the triangle drawn to the base.

∆АВС – isosceles. AB = BC = 13 cm.

Since the perimeter P = AB + BC + AC = 36 cm, then AC = 36 – 13 – 13 = 10 cm.

The median in a triangle bisects the opposite side. So AH = CH = 5 cm.

The median in an isosceles triangle is the height. Hence it follows that ∆АВН is rectangular.

By the Pythagorean theorem, we have AB ^ 2 = AH ^ 2 + BH ^ 2 or BH = √ (AB ^ 2 – AH ^ 2) = √ (13 ^ 2 – 5 ^ 2) = √144 = 12 cm.

Answer: AN = 12 cm.