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In figure, PQ is a chord of a circle with center O and PT is a tangent. If $∠ QPT = 60 °$ , find $∠ PRQ .$

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150 o

120 o

60 o

180 o

answer is B.

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Detailed Solution

Given that PQ is a chord of a circle with center O and PT is a tangent.
We have to find the

∠ PRQ .

The tangent of a circle at any point in a circle is perpendicular to the radius through the point of contact.
From the figure, PT is the tangent and OP is the radius. So,

∠ O P T = 90 °

Thus,

∠ O P Q = ∠ O P T − ∠ Q P T ⇒ ∠ O P Q = 90 ° − 60 ° ⇒ ∠ O P Q = 30 °

In the triangle,

O P Q, O P = O Q

because of radii of the same circle. So,

∠ O Q P = ∠ O P Q = 30 °

The sum of the three angles of a triangle is equal to 180 degrees. So,

∠ O Q P + ∠ O P Q + ∠ P O Q = 180 ° ⇒ 30 ° + 30 ° + ∠ P O Q = 180 ° ⇒ ∠ P O Q = 180 ° − 60 ° ⇒ ∠ P O Q = 120 °

Therefore, the angle formed by the major arc is,

∠ P O Q = 360 ° − 120 ° ∴ ∠ P O Q = 240 °

The angle subtended by an arc at center is double the angle subtended by it on the remaining part of the circle Thus,

∠ P R Q = 12 ∠ P O Q ⇒ ∠ P R Q = 12 240 ° ⇒ ∠ P R Q = 120 °

The value of

∠ P R O

120 o

.
Therefore, the correct option is 2.

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