In the figure above, AB is a tangent to the circle at P. Is PQ a radiu

Question

https://gmatclub.com/forum/download/file.php?id=63717 In the figure above, AB is a tangent to the circle at P. Is PQ a radius of the circle? (1) ∠QPA is a right angle (2) Q is the center of the circle Project DS Butler Data Sufficiency (DS3) For DS butler Questions Click Here 1.png

manish1708 · Answer

IMO: B

Given: AB is a tangent to the circle at P.

Statement 1: ∠QPA is a right angle.
But we do not know if Q is center of the circle or not.
so, PQ can or cannot be the radius.
Hence, insufficient.

Statement 2: Q is the center of the circle
and we know radius is center of circle to the point of tangent.
so, PQ is radius of the circle.
Hence, sufficient.

100mitra · Answer

Correct Option B

Statement 1: not sufficient, no details about Q position in circle

Statement 2 : sufficient, Q center point joins tangent perpendicular to circle at point P. Makes it radius of circle.

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reviewtattantat · Answer

AB is a tangent to the circle at P. 
(1) ∠QPA is a right angle  => No clue where Q is => insufficient

(2) Q is the center of the circle => sufficient

B

Izziedodat · Answer

So my question for the first statement is based on the theorem of tangent lines. A tangent line can only be if it is perpendicular to the radius at which forms a right triangle. If we know that the tangent line forms a right angle, why cant statement 1 be sufficient?

thhegeshwar · Answer

If AB is tangent to the circle at P then angQPA should be the only 90deg right? Because Q is at the centre of the circle.
Am i missing something?
Because as far as i know only the point on the centre of the circle can make a 90deg with the tangent point

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Victor19 · Answer

The answer is B.

To determine if QP is the radius of the circle, we need to position of Q as P is already a point at the circumference and AB is tangent at P.

Statement 1 : Point Q could be anywhere along the perpendicular line to AB.
Not Sufficient.

Statement 2 : If Q is the center of the circle and P is a point at the circumference, then QP MUST be the radius.
Sufficient.

EnthuCutlet99 · Answer

As per circle properties, a line within a circle that is perpendicular to a tangent at the point of tangency has to be the radius of the radius of the circle. Also, the radius of the circle has to, by definition, go through the center of the circle. In this case, shouldn't the answer be D? Am I missing something

Victor19 · Answer

You are partially right. Where you made a mistake is when you said "at the point of tangency has to be the radius of the radius of the circle". Actually, at the point of tangency the perpendicular has to pass through the center, but for the perpendicular to be the radius the line should terminate at the center. Even the diameter is perpendicular at the tangent. Or Q could be midway between the P and the center of the circle. So you can never be sure.

That is why "A" is not an answer and hence "D" is also not an answer.

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In the figure above, AB is a tangent to the circle at P. Is PQ a radiu

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In the figure above, AB is a tangent to the circle at P. Is PQ a radiu

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