Question

PQ is tangent to a circle with centre O. If \(\angle POR = 65^\circ\), then m\(\angle PTR\) is

• A. \(65^\circ\)
• B. \(58.5^\circ\)
• C. \(57.5^\circ\)
• D. \(45^\circ\)

Hint:

In any right-angled triangle like \(\Delta SPT\), the two acute angles must add up to \(90^\circ\).
Once you find the inscribed angle \(\angle PST = 32.5^\circ\), you can simply subtract it from \(90^\circ\) to find the answer:
\[ \angle PTR = 90^\circ - 32.5^\circ = 57.5^\circ \] This shortcut avoids working with \(180^\circ\) and saves valuable time!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The topic of this question is Circles and Geometry.
We are given a circle centered at \(O\) where \(PQ\) is a tangent to the circle.
The angle subtended by the arc \(PR\) at the center of the circle is \(\angle POR = 65^\circ\).
We need to determine the measure of the angle \(\angle PTR\).

Step 2: Key Formula or Approach:
1. The angle subtended by an arc of a circle at the center is double the angle subtended by the same arc at any point on the remaining part of the circle.
2. The radius drawn to the point of tangency is perpendicular to the tangent line:
\[ \angle SPT = 90^\circ \] 3. The sum of the interior angles of a triangle is always \(180^\circ\).

Step 3: Detailed Explanation:

• Identify the central angle subtended by the arc \(PR\):
\[ \angle POR = 65^\circ \]

• Find the angle subtended by arc \(PR\) at point \(S\) on the circumference:
According to the central angle theorem, \(\angle PSR\) is half of \(\angle POR\):
\[ \angle PSR = \frac{1}{2} \angle POR = \frac{65^\circ}{2} = 32.5^\circ \]

• Identify the perpendicular relationship:
\(SP\) is the diameter of the circle, and \(PQ\) is the tangent at point \(P\).
Therefore, the radius \(OP\) (and hence the diameter \(SP\)) is perpendicular to the tangent line at \(P\):
\[ \angle SPT = 90^\circ \]

• Consider the right-angled triangle \(\Delta SPT\):
The three points \(S\), \(R\), and \(T\) lie on a straight line.
The interior angles of \(\Delta SPT\) must add up to \(180^\circ\):
\[ \angle PST + \angle SPT + \angle PTS = 180^\circ \] Substitute the known values \(\angle PST = 32.5^\circ\) and \(\angle SPT = 90^\circ\) into the equation:
\[ 32.5^\circ + 90^\circ + \angle PTR = 180^\circ \] \[ 122.5^\circ + \angle PTR = 180^\circ \] \[ \angle PTR = 180^\circ - 122.5^\circ = 57.5^\circ \]

Step 4: Final Answer:
The measure of \(\angle PTR\) is \(57.5^\circ\).
Therefore, the correct option is (C).