Given: ∆ABC is isosceles
m∠ACB = 120°
m∠BMC = 60°
CM = 12
Find: AB
can you guys help I'm really struggling with proofs (statement and reasons)
Sketch is the big thing!!
On mine , I extended AC to M so that angle BMC = 60°
Also you can see that angle MCB = 60° (external angle)
So triangle CBM is equilater, and since CM = 12
CB = MB = 12
Also, since ABC is isosceles and CA = CB
CA = 12
now by the sine law:
AB/sin120° = 12/sin30
AB = ...
To find the length of AB, we need to use the properties of the given isosceles triangle ∆ABC and the given angle measures.
First, let's draw a diagram of the given information. We have an isosceles triangle ∆ABC, where AC = BC, and m∠ACB = 120°. We also have a point M on side AC, and m∠BMC = 60°, and CM = 12.
Now, let's analyze the given information and use some properties of isosceles triangles to solve for AB.
1. Property of Isosceles Triangles: In an isosceles triangle, the angles opposite the congruent sides are congruent.
Since ∆ABC is isosceles and AC = BC, we can conclude that m∠ABC = m∠ACB = 120°.
2. Property of Triangle Angles: The sum of the angles in a triangle is 180°.
We can use this to find m∠BAC:
m∠BAC + m∠ABC + m∠ACB = 180°
m∠BAC + 120° + 120° = 180°
m∠BAC = 180° - 240°
m∠BAC = -60°
3. Property of Triangle Angles: All angles in a triangle must be greater than 0°.
We can't have a negative angle in a triangle, so we discard the value of m∠BAC = -60°.
4. Property of Triangle Sides: The sum of the lengths of any two sides of a triangle is always greater than the length of the third side (Triangle Inequality).
Let's apply the Triangle Inequality to ∆BMC:
BM + CM > BC
BM + 12 > BC
BM > BC - 12
We know that BC = AC because ∆ABC is isosceles, so we can rewrite the inequality as:
BM > AC - 12
BM > AB - 12
5. Property of Triangle Angles: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Using this property, we can find m∠BMC:
m∠BMC = m∠ACB + m∠ABC
60° = 120° + m∠ABC
m∠ABC = 60° - 120°
m∠ABC = -60°
6. Property of Triangle Angles: All angles in a triangle must be greater than 0°.
We can't have a negative angle in a triangle, so we discard the value of m∠ABC = -60°.
7. Property of Triangle Sides: The lengths of the sides opposite congruent angles in an isosceles triangle are congruent.
Since m∠ACB = m∠ABC = 120°, we can conclude that BM = AB.
Now, let's go back to the previous inequality:
BM > AB - 12
Since BM = AB in an isosceles triangle, we can replace BM with AB:
AB > AB - 12
This inequality tells us that AB must be greater than AB - 12, which means that the length of AB must be greater than 12.
Therefore, the length of AB is greater than 12, but we can't determine the exact length without more information or measurements.