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Mathematics — CBSE Class 10 board question

Q1. [1]
In the given figure, PT is a tangent to the circle with centre O and radius r. If $\angle POT = 45°$, then the length of OP is :
  1. A $r\sqrt{2}$
  2. B $\sqrt{2r}$
  3. C $2r$
  4. D $r^2$
Previously asked in CBSE board exam
2026 30/3/1 Q12
Generated by claude-sonnet-4-6 · 2026-06-15 10:26 · grounding rag
Model Answer

Option A: $r\sqrt{2}$

Since PT is a tangent at T, $\angle OTP = 90°$. In right triangle OTP, $\angle POT = 45°$, so $\cos 45° = \dfrac{OT}{OP} = \dfrac{r}{OP}$, giving $OP = \dfrac{r}{\cos 45°} = r\sqrt{2}$.

Explanation

The key property used is: tangent ⊥ radius at point of contact (Theorem 10.1), so ∠OTP = 90°. Then apply trigonometry (or Pythagoras) in right triangle OTP. With ∠POT = 45° and the remaining angle ∠TPO = 45°, the triangle is an isosceles right triangle, so $OP = OT\sqrt{2} = r\sqrt{2}$. Watch out for option B ($\sqrt{2r}$) — it is a common distractor; the correct form is $r\sqrt{2}$, not $\sqrt{2r}$.

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