Question

In the figure below, PT and ST are two secants. If O is the centre of the circle and \( PQ = 2QT = 8 \) cm, \( OS = 5 \) cm, then what is the measure of the line OT? (Figure not drawn to scale)

• A. \( \sqrt{54} \) cm
• B. \( \sqrt{60} \) cm
• C. 8 cm
• D. \( \sqrt{73} \) cm

Hint:

Use OS as the radius, then apply the power of a point theorem on the secant through P and Q.

Answer 1

The Correct Option is D

Step 1: Note the known lengths.
S lies on the circle, so OS is a radius, giving \( r = 5 \) cm.
\( PQ = 8 \) cm and \( QT = 4 \) cm, since \( PQ = 2QT = 8 \).

Step 2: Find the full secant length PT.
P and Q both lie on the circle, and Q sits between P and T on the secant line.
So \( PT = PQ + QT = 8 + 4 = 12 \) cm.

Step 3: Apply the power of a point.
For an external point T, the power along secant PQT equals \( TQ \times TP \).
\( TQ \times TP = 4 \times 12 = 48 \).

Step 4: Link the power to OT.
For any external point, power also equals \( OT^2 - r^2 \).
So \( OT^2 - 25 = 48 \), which gives \( OT^2 = 73 \).

Final Answer:
\( OT = \sqrt{73} \) cm. \[ \boxed{OT = \sqrt{73} \text{ cm}} \]