In a circle, chord AB subtends an angle of 75° at the circumference at point C. A tangent is drawn at point A. If the angle between this tangent and chord AC is 40°, what is the measure of the angle subtended by the minor arc AC at the center of the circle?
Show answer & explanation
Correct answer: A
Explanation
Given:
∠ACB = 75°
Angle between tangent and chord AC = 40°
Formula:
Tangent-Chord Theorem:
Angle between tangent and chord = Angle subtended by the opposite arc on the circumference
Angle subtended at the center = 2 × Angle subtended at the circumference
∠ABC = 40°
∠ACB = 75°
In triangle ABC,
∠BAC = 180° − (75° + 40°) = 65°
Angle subtended by arc AC on the circumference = ∠ABC = 40°
Angle subtended at the center = 2 × 40° = 80°
Angle subtended by minor arc AC at the center = 80°