Question

In the given figure, \(\overline{PQ}\) is the diameter of a circle with center \(O\). Two points \(R\) and \(S\) are chosen on the circle such that \(\angle ROS = 80^\circ\). When \(\overline{PR}\) and \(\overline{QS}\) are extended, they meet at \(T\).

The value of \(\angle RTS\) is ______

• A. 40°
• B. 50°
• C. 60°
• D. 80°

Hint:

Use the rule that an inscribed angle is half the central angle to find angle RQS first.
Then use the right angle in the semicircle and the angle sum of triangle TRQ.

Answer 1

The Correct Option is B

Step 1: Understand the question:
PQ is a diameter of the circle with center O, and R, S sit on the circle with central angle ROS equal to 80 degrees.
Lines PR and QS are extended outward and meet at point T, and we need to find the angle RTS formed at that meeting point.

Step 2: Use the central and inscribed angle rule:
The angle an arc makes at the center is always double the angle it makes at any other point on the circle.
Arc RS makes a central angle of 80 degrees at O, so it makes half of that, 40 degrees, at Q. This gives angle RQS = 40 degrees.

Step 3: Use the angle in a semicircle:
Since PQ is a diameter, angle PRQ must be a right angle, so angle PRQ = 90 degrees.
Point T lies on the extension of PR beyond R, so angle TRQ is the supplement of angle PRQ along the straight line PRT, giving angle TRQ = 180 - 90 = 90 degrees.

Step 4: Solve the triangle TRQ:
In triangle TRQ the three angles add up to 180 degrees, and angle RQT = angle RQS = 40 degrees since Q, S, T lie on one line.
\[ \angle RTS + 90^\circ + 40^\circ = 180^\circ \]
\[ \angle RTS = 180^\circ - 130^\circ = 50^\circ \]

Step 5: Check option (A) 40 degrees.
40 degrees is the value of angle RQS found in Step 2, not the final angle RTS, so this option is wrong.

Step 6: Check option (B) 50 degrees.
This matches the value calculated directly in Step 4, so this option is correct.

Step 7: Check option (C) 60 degrees.
60 degrees does not satisfy the triangle angle sum with 90 and 40 degrees, so this option is wrong.

Step 8: Check option (D) 80 degrees.
80 degrees is just the given central angle ROS repeated, not the angle at T, so this option is wrong.

Final Answer:
The angle RTS works out to 50 degrees. \[ \boxed{\angle RTS = 50^\circ} \]