Given: ∆ABC is isosceles m∠ACB = 120° M ∈ AB , CM = 12 m∠BMC = 60° Find: AB

Look at the picture.This is isosceles triangle. Therefore, the angles at the base are congruent.m∠ BAC = m∠ABC.We know, the sum of the measures of the angles in a triangle is equal 180°.Therefore:m∠BAC + m∠ABC + m∠ACB = 180°2m∠ABC + 120° = 180°     subtract 120° from both sides2m∠ABC = 60°     divide both sides by 2m∠ABC = 30°m∠MCB = 90° → ΔMCB is a right triangle (30 - 60 - 90).The sides of that triangle are in proportion: 1 : √3 : 2.Therefore MC : BC : MB = 1 : √3 : 2MC = 12 → BC = 12√3 and MB = 2 · 12 = 24ΔCDB is the right triangle (30 - 60 - 90) too.Therefore CD : BD : BC = 1 : √3 : 2CD = 1/2 BC → CD = 1/2(12√3) = 6√3BD = CD√3 → BD = 6√3(√3) = 6(3) = 18AB = 2BD therefore your answer is:AB = 2(18) = 36