Given: ∆ABC is isosceles
m∠ACB = 120°
m∠BMC = 60°
CM = 12
Find: AB
M is on line segment AB
To find the length of AB, we need to use the properties of an isosceles triangle and the angle bisector theorem.
First, let's draw the triangle ∆ABC. Since it is an isosceles triangle, this means that AB and AC are congruent. Hence, we can label AB as x and AC as x as well.
Next, let's denote the point where the angle bisector of ∠ACB intersects side AB as M. We are given that ∠ACB = 120°, ∠BMC = 60° and CM = 12.
Since ∠BMC is an interior angle of ∆ABC, we can use the angle sum property of a triangle to find ∠AMB. The sum of all angles in a triangle is 180°, so ∠AMB = 180° - ∠BMC - ∠ACB = 180° - 60° - 120° = 0°.
Now, let's apply the angle bisector theorem. According to the theorem, the ratio of the lengths of the two segments formed by the angle bisector is equal to the ratio of the lengths of the other two sides of the triangle. In other words, AM/MB = AC/CB.
Since AM = x (because it is part of side AB) and AC = x (because it is congruent to AB), we can rewrite the equation as x/MB = x/x. Simplifying this, we get 1 = MB/x.
From this equation, we can conclude that MB = x.
Since ∠AMB = 0°, this means that M is collinear with A and B. Hence, MB can be the length of AB.
Therefore, we can conclude that AB = MB = x.
Now, we need to find the value of x.
Since ∆ABC is an isosceles triangle, we have two equal angles, ∠ACB and ∠ABC. Therefore, we can use the fact that the sum of the angles in a triangle is 180° to find ∠ABC.
∠ACB + ∠ABC + ∠CAB = 180°
120° + ∠ABC + ∠ABC = 180° (Using the fact that ∠ACB = 120°)
2∠ABC = 180° - 120°
2∠ABC = 60°
∠ABC = 60°/2
∠ABC = 30°
So, ∠ABC = 30°.
Since ∆ABC is isosceles, we know that the base angles, ∠ABC and ∠ACB, are congruent. Therefore, ∠ACB is also 30°.
Now, let's use the trigonometric properties to find x.
In ∆ACM, we have the angle ∠ACM = 30°, the opposite side CM = 12, and we need to find the adjacent side AM.
Using the trigonometric function cosine, we can write:
cos(30°) = AM / CM
1/2 = AM / 12
AM = 12/2
AM = 6
Since AM = x, we can conclude that x = 6.
Therefore, the length of AB is 6 units.
surely by now you could have figured this out.
Since ∆ABC is isosceles, ∠A = ∠B = 30°
Therefore, ∆BMC is a 30-60-90 right triangle, with ∠MCB = 90°
So, BC = 12√3
Now drop an altitude from C to AB and use BC to find AB.