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: Set up the secant angle rule:
When two secants are drawn from an outside point, the angle between them equals half the difference between the far arc and the near arc they cut off.
Here the near arc is arc RS and the far arc is arc PQ on the side away from R and S.

Step 2: Find the near arc.
The near arc RS is given directly by the central angle, so arc RS $= 80^\circ$.

Step 3: Find the far arc.
Since PQ is a diameter, it splits the circle into two arcs of $180^\circ$ each, so the far arc PQ on the side away from R and S is one full semicircle, $180^\circ$.

Step 4: Apply the formula.
\[ \angle RTS = \frac{\text{far arc} - \text{near arc}}{2} = \frac{180^\circ - 80^\circ}{2} = 50^\circ \]

Step 5: Check option (A) 40 degrees.
40 degrees would come from taking half the near arc alone, $80^\circ / 2$, without subtracting from the far arc, so it is wrong.

Step 6: Check option (B) 50 degrees.
This is exactly what the secant formula gives in Step 4, so it is the right option.

Step 7: Check option (C) 60 degrees.
60 degrees does not fit the formula for any reasonable pairing of the arcs here, so it is wrong.

Step 8: Check option (D) 80 degrees.
80 degrees is just the near arc value copied over without applying the formula, so it is wrong.

Final Answer:
Using the exterior angle formula for two secants also gives 50 degrees. \[ \boxed{\angle RTS = 50^\circ} \]