Mathematics
The figure shows two circles which intersect at A and B. The center of the smaller circle is O and lies on the circumference of the larger circle. Given ∠APB = a°.
Calculate, in terms of a°, the value of :
(i) obtuse ∠AOB,
(ii) ∠ACB,
(iii) ∠ADB.
Give reasons for your answers clearly.
Answer
(i) We know that,
Angle at the center is double the angle at the circumference subtended by the same chord.
obtuse ∠AOB = 2∠APB = 2a°.
Hence, obtuse ∠AOB = 2a°.
(ii) OACB is a cyclic quadrilateral.
⇒ ∠AOB + ∠ACB = 180° [Sum of opposite angles in a cyclic quadrilateral = 180°.]
⇒ ∠ACB + 2a° = 180°
⇒ ∠ACB = 180° - 2a°.
Hence, ∠ACB = 180° - 2a°.
(iii) Join AD and BD.
As, angles in same segment are equal.
∴ ∠ADB = ∠ACB = 180° - 2a°.
Hence, ∠ADB = 180° - 2a°.
Related Questions
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(ii) ∠BDC.
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