The centre of the circle passing through the midpoints of the sides of an isosceles triangle
ABC lies on the circumcircle of triangle ABC. If the larger angle of triangle ABC is Alphaº and the
smaller one Betaº then what is the value of Alpha- Beta?
Please explain in copy with diagram.
I will definitely mark as brainliest if I can able to understand the answer.
Answers
Given : The centre of the circle passing through the midpoints of the sides of an isosceles triangle ABC lies on the circumcircle of triangle ABC.
larger angle of triangle ABC is α and the smaller one β
To Find : what is the value of α - β
-
Solution:
Let say AC = BC = 2x ( equal sides)
and AB = 3rd side
A circle passing through mid point of AB & BC Hence center of circle will lie on perpendicular bisector of AB
As center of circle lies on circumcircle hence only possible center would point C
Hence AC/2 = BC/2 = 2x/2 = x
and CM = AC/2 = BC/2 = x
and CM ⊥ AB
Sin ∠CAM = CM/AC
∠CAM = ∠CAB
=> Sin ∠CAB = x/2x
=> Sin ∠CAB = 1/2
=> ∠CAB = 30° can not be 150° as ∠CBA = ∠CAB
∠ CBA = 30°
∠ACB = 180° - ∠CAB - ∠ CBA
=> ∠ACB = 180° - 30° - 30°
=> ∠ACB = 120°
∠ACB = 120° = α larger angle
∠CAB = ∠ CBA = 30° = β smaller angle
α - β = 120° - 30° = 90°
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