Book Online Demo

Check Your IQ

Try Test

Courses

Offline Centres

In figure, AB is a chord of a circle with centre O and AP is the tangent at A such that $∠ BAP = 75^{\circ}$ . Then, which of the following is the measure of $∠ ACB$ ?

see full answer

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

135^{\circ}

120^{\circ}

105^{\circ}

90^{\circ}

answer is C.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Given,
AB is a chord of a circle with centre O and AP is the tangent at A .

∠ B A P = 75^{\circ}

Here,

∴ ∠ OAP = 90^{\circ}

∠ OAP = ∠ OAB + ∠ BAP

∠ OAB = ∠ OAP - ∠ BAP

∴ ∠ OAB = 90^{\circ} - 75^{\circ}

∠ OAB = 15^{\circ}

In triangle OAB,
By angle sum property,

∠ AOB + ∠ OAB + ∠ OBA = 180 °

We know that,

∠ OAB = ∠ OBA

[since OA=OB(Radius)]
So,

∠ AOB = 180^{°} - (∠ OAB + ∠ OBA)

∠ AOB = 180^{°} - 2 ∠ OAB

∠ AOB = 180^{°} - 2 (15^{°})

∠ AOB = 150^{°}

∴

Exterior

∠ AOB = 360^{°} - 150^{°}

Exterior

∠ AOB = 210^{°}

So,

∠ ACB = \frac{1}{2} \times 210^{°}

∠ ACB = 105^{°}

[Since angle subtended by an arc at the centre is double than the angle subtended by it at the remaining part of the circle]
Hence, the measure of

∠ ACB

105^{°}

.
Therefore, option 3 is correct.

Watch 3-min video & get full concept clarity