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In figure, O is the centre of the circle. Determine and $∠$ AOB if PA and PB are tangents and $∠$ ABP $= 75 °$ .

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∠

ABP

= 52.5 °

and

∠

ABP

= 127.5 °

∠

ABP

= 50.5 °

and

∠

ABP

= 127.5 °

∠

ABP

= 52.5 °

and

∠

ABP

= 128.5 °

∠

ABP

= 52.5 °

and

∠

ABP

= 120.5 °

answer is A.

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Detailed Solution

Given that,

∠

APB = 75°.
Question Image

PA and PB are the tangents to the circle from P at A and B respectively and OA and OB are the radii of the circle.
Hence, OA _|_ PA and OB_|_ PB.
Now,

∠

OAP =

∠

OBP = 90°.
In quadrilateral AOBP,
Sum of all the angles of a quadrilateral is 360°.

∠

ABP +

∠

OAP +

∠

OBP +

∠

AOB = 360°

\Rightarrow

75° + 90° + 90° +

∠

AOB = 360°

\Rightarrow

255° +

∠

AOB = 360°

∠

AOB = 360° - 255°

∠

AOB = 105°
We know that angle subtended by an arc at the centre is double the angle subtended by the same arc at any point on the circle. Consider the minor arc AB, then