Question:medium

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)

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Use OS as the radius, then apply the power of a point theorem on the secant through P and Q.
Updated On: Jul 21, 2026
  • \( \sqrt{54} \) cm
  • \( \sqrt{60} \) cm
  • 8 cm
  • \( \sqrt{73} \) cm
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The Correct Option is D

Solution and Explanation

Step 1: Mark the midpoint of chord PQ.
Since P and Q both sit on the circle, PQ is a chord of length 8 cm.
Let M be the midpoint, so $ PM = QM = 4 $ cm. The perpendicular from the centre O to a chord always bisects it, so $ OM \perp PT $.

Step 2: Find OM using the radius.
In right triangle OMQ, $ OQ = r = 5 $ cm (radius) and $ QM = 4 $ cm.
$ OM^2 = OQ^2 - QM^2 = 25 - 16 = 9 $, so $ OM = 3 $ cm.

Step 3: Find the distance MT along the secant.
M to Q is 4 cm, and Q to T is another 4 cm, both along the same straight line PT.
So $ MT = MQ + QT = 4 + 4 = 8 $ cm.

Step 4: Use the right triangle OMT.
Since $ OM \perp MT $, apply the Pythagorean theorem.
$ OT^2 = OM^2 + MT^2 = 9 + 64 = 73 $.

Final Answer:
$ OT = \sqrt{73} $ cm, matching the power of a point result. $ OT = \sqrt{73} \text{ cm} $
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