Question
Download Solution PDFIn the figure given below, DE is a tangent to the circle with centre O. If ∠ACD = 50° and ∠AOB = 140°, find the value of ∠OAC.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
DE is a tangent to the circle with centre O
∠ACD = 50°
∠AOB = 140°
Concept:
Theorem∶ The angle subtended by an arc at the centre of the circle is double the angle subtended by it at the circumference of the circle.
Theorem∶ According to the alternate segment theorem, the angle between a tangent and a chord is equal to the angle made by the chord in the alternate segment of the circle.
Calculation:
According to the question:
∵ OA = OB = radius of circle
⇒ ΔAOB is an isosceles triangle.
⇒ ∠OAB = ∠OBA
∵ Sum of angles of triangle = 180°
⇒ ∠OAB + ∠OBA + ∠AOB = 180°
⇒ ∠OAB = ∠OBA = (180° – 140°)/2 = 20°
According to the concept,
⇒ ∠ACB = 1/2 × ∠AOB = 140°/2 = 70°
⇒ ∠BCE = 180° – ∠ACB – ∠ACD = 180° – 70° – 50° = 60°
Again, according to the concept,
⇒ ∠BCE = ∠BAC = 60°
⇒ ∠OAC = ∠BAC – ∠OAB = 60° – 20° = 40°
∴ The required result will be 40°.
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