In an isosceles triangle ABC, height BD = 12 cm, lateral side = 13 cm. Find side AC and height AK.

Since the triangle ABC, by condition, is isosceles, then the height BD will also be its median, then AD = CD, and AC = 2 * AD.

In a right-angled triangle ABD, according to the Pythagorean theorem, we determine the length of the leg AD.

AD ^ 2 = AB ^ 2 – BD ^ 2 = 169 = 144 = 25.

AD = 5 cm, then AC = 2 * AD = 2 * 5 = 10 cm.

Determine the area of the triangle ABC.

Savs = BD * AC / 2 = 12 * 10/2 = 60 cm2.

Also, the area of the triangle ABC is equal to: Savs = BC * AK / 2.

AK = 2 * Savs / BC = 2 * 60/13 = 120/13 = 9 (3/13) cm.

Answer: The length of the base of the AC is 10 cm, the height of the AC is 9 (3/13) cm.