In a circle with centre O, PQ is a chord. A line segment ROS is drawn from a point R on the circle through the centre O to meet the extension of chord PQ at point S, such that QS = OR. If angle QSR = 30°, then what is the measure (in degrees) of angle POR?

Introduction / Context:
This is a challenging circle geometry question that combines chords, radii, and special angle constructions. A line passes through a point on the circle and the centre, then meets the extension of a chord such that a segment on the extension is equal in length to the radius. An angle at this external point is given, and we must determine a central angle inside the circle.

Given Data / Assumptions:

• O is the centre of the circle.
• P and Q are points on the circle, and PQ is a chord.
• R is a point on the circle.
• Line segment ROS passes through R and O and meets the extension of chord PQ at S.
• QS = OR, and OR is a radius of the circle.
• Angle QSR at the external point S is 30°.
• We must find angle POR, the angle at the centre between OP and OR.

Concept / Approach:
The key ideas are symmetry and chord-line intersection geometry. By placing the circle conveniently in a coordinate system and using the condition QS = radius, we can determine the specific configuration that satisfies all constraints. Once the coordinates of P, Q and R are known, the angle at the centre between OP and OR can be computed. The configuration leads to a right angle at the centre between OP and OR.

Step-by-Step Solution:
Step 1: Without loss of generality, consider a circle of radius 1 centred at the origin O(0, 0) in the coordinate plane.Step 2: Place point R at (1, 0) on the circle, so OR is along the positive x axis.Step 3: Since ROS is a straight line through R and O, point S lies on the extension of this line beyond O, so S has coordinates (-t, 0) for some t > 0.Step 4: Let the line through S that meets the circle at P and Q have slope m. Because angle QSR is 30° and SR lies along the positive x axis, the line SQ makes an angle of 30° with the x axis, so its slope m = tan 30° = 1 / √3.Step 5: The equation of line SQ is therefore y = (1 / √3)(x + t), passing through S(-t, 0).Step 6: Find intersections of this line with the circle x^2 + y^2 = 1. Solving simultaneously yields two points: Q and P. One of these points is closer to S and one is further; the nearest intersection to S is Q.Step 7: The distance condition QS = OR means that QS must equal 1, the radius. Imposing this distance and solving for t gives t = √3. At this value, the intersection points are Q(-√3/2, 1/2) and P(0, 1).Step 8: With these coordinates, OP is the vector from O to P(0, 1), which lies along the positive y axis, and OR is the vector from O to R(1, 0), which lies along the positive x axis.Step 9: The angle between OP and OR is simply the angle between the x and y axes, which is 90°. Thus angle POR = 90°.

Verification / Alternative check:
A purely geometric verification is possible by observing that the construction yields an isosceles right triangle at the centre, due to the symmetric placement of points and equal segment condition QS = OR. This symmetry implies that the central angle between OP and OR is a right angle, confirming the computed result of 90°.

Why Other Options Are Wrong:
Angles 30°, 45° and 60° typically arise from equilateral and simple special right triangle relations, but they do not satisfy the specific distance condition QS = radius together with the given 30° external angle. An obtuse angle such as 120° does not match the perpendicular relationship between OP and OR that is forced by the configuration.

Common Pitfalls:
This question can be confusing because it mixes an external angle and a length condition that involves both a chord and a radius. A common mistake is to assume that the angle at the centre directly equals twice the angle at the circumference, which does not apply in this external configuration. Another error is to ignore the distance condition QS = OR and try to guess the angle from standard circle theorems alone. Using a clear analytic or carefully reasoned geometric approach avoids these traps.

Final Answer:
The measure of angle POR is 90°.