Mathematics
In the figure (i) given below, AB is a chord of the circle with centre O, BT is tangent to the circle. If ∠OAB = 32°, find the values of x and y.
Circles
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Answer
In △OAB,
OA = OB (∵ both are radius of the common circle.)
So, △OAB is a isosceles triangle with,
∠OBA = ∠OAB = 32°.
Since sum of angles in a triangle = 180°.
In △OAB,
⇒ ∠OBA + ∠OAB + ∠AOB = 180°
⇒ 32° + 32° + ∠AOB = 180°
⇒ 64° + ∠AOB = 180°
⇒ ∠AOB = 180° - 64°
⇒ ∠AOB = 116°.
Arc AB subtends ∠AOB at centre and ∠ACB at remaining part of circle.
∴ ∠AOB = 2∠ACB (∵ angle subtended at centre is double the angle subtended at remaining part of the circle.)
⇒ 116° = 2y
⇒ y =
⇒ y = 58°.
From figure,
∠ABT = ∠ACB = 58° (∵ angles in alternate segments are equal.)
∴ x = 58°.
Hence, the value of x = 58° and y = 58°.
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