Question
Download Solution PDFAB is a chord of a circle with centre O and P is any point on the circle. If ∠APB = 122°, then what is the measure of ∠OAB ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
∠APB = 122°
Concept used:
Sum of opposite sides of a cyclic quadrilateral are 180°
The angle subtended by an arc of a circle at its center is twice the angle it subtends anywhere on the circle’s circumference.
Calculation:
In the given figure,
APBT is a cyclic quadrilateral.
∠ATB + ∠APB = 180° [sum of opposite angles are 180°]
⇒ x° + 122° = 180°
⇒ x = (180° – 122°)
⇒ x = 58°
∠ATB = 58°
And we know that the angle subtended by an arc of a circle at its center is twice the angle it subtends anywhere on the circle’s circumference.
⇒ ∠AOB = 2 × ∠ATB
⇒ ∠AOB = 2 × 58°
⇒ ∠AOB = 116°
Now,
OA = OB [radius of the circle]
Then,
∠OAB + ∠AOB + ∠OBA = 180°
⇒ θ + 116° + θ = 180°
⇒ 2θ + 116° = 180°
⇒ 2θ = (180° – 116°)
⇒ 2θ = 64°
⇒ θ = 32°
so,
∠OAB = θ = 32°
∴ The required value of ∠OAB is 32°.
Shortcut Trick
From the above given diagram we have
⇒ θ = P - 90°
⇒ θ = 122° - 90° = 32°
∴ The correct answer is 32°.
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