In an isosceles triangle, the median to the base is 16 cm, and the median to the side is 2√97 cm

In an isosceles triangle, the median to the base is 16 cm, and the median to the side is 2√97 cm. Find the perimeter of the triangle.

Given:
AB = BC
BD = 16cm
AE = 2√97 cm

To find:
P (ABC) -?

1) The medians of an isosceles triangle at intersection are divided in a ratio of 2: 1, counting from the top;
2) OD = 1/3 * BD = 1/3 * 16 = 16/3 (cm);
3) AO = 2/3 * AE = 2/3 * 2√97 = (4/3) √97 (cm);
4) AD = √ (AO ^ 2 – OD ^ 2) = √ (((4/3) √97) ^ 2 – (16/3) ^ 2) = 12 (cm);
5) AC = 2 * AD = 2 * 12 = 24 (cm);
6) AB = √ (AD ^ 2 + BD ^ 2) = √ (12 ^ 2 + 16 ^ 2) = 20 (cm);
7) P (ABC) = AB + BC + AC = 20 + 20 + 24 = 64 (cm).
Answer: The perimeter of ABC is 64 cm.