In the given figure, 'O' is the centre of the circle. Determine ∠AQB and ∠AMB , if PA and PB are tangents and ∠APB= 75 ° .

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A
${50.5}^{\circ}$ and ${125.5}^{\circ}$ respectively
B
${125.5}^{\circ}$ and ${50.5}^{\circ}$ respectively
C
${52.5}^{\circ}$ and ${127.5}^{\circ}$ respectively
D
${127.5}^{\circ}$ and ${52.5}^{\circ}$ respectively

Solution:

Given that

∠ A P B = 75 °

To find :

∠ A M B

and

∠ A Q B

As tangents are perpendicular to radius. Thus,

∠ P A O = 90 ° ∠ P B O = 90 °

Using angle sum property of quadrilaterals in quad. APBO , find

∠ A O B

∠ AOB + ∠ PAO + ∠ PBO + ∠ APB = 360^{⚬}

\Rightarrow ∠ AOB = 360^{⚬} - (∠ PAO + ∠ PBO + ∠ APB)

\Rightarrow ∠ AOB = 360^{⚬} - (75^{⚬} + 90^{⚬} + 90^{⚬})

\Rightarrow ∠ AOB = 105^{⚬}

Since, the angle subtended by an arc at the centre is double the angle subtended at any point on the circle. Thus,

∠ A O B = 2 ∠ A Q B

.
Now find

∠ A Q B

\Rightarrow ∠ AQB = \frac{105^{\circ}}{2}

\Rightarrow ∠ AQB = {52.5}^{\circ}

AQBM is a cyclic quadrilateral, find

∠ A M B

∠ AQB + ∠ AMB = 180^{\circ}

\Rightarrow ∠ AMB = 180^{\circ} - ∠ AQB

\Rightarrow ∠ AMB = 180^{\circ} - {52.5}^{\circ}

\Rightarrow ∠ AMB = {127.5}^{\circ}

Hence, measure of

∠ A M B

127.5 °

and

∠ AQB

{52.5}^{\circ}

.
Therefore, option 4 is correct.

In the given figure, 'O' is the centre of the circle. Determine $∠ AQB$ and $∠ AMB$ , if $PA$ and $PB$ are tangents and $∠ APB = 75 °$ .

Solution:

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