CBSE Class 10 Mathematics Basic 2024: question 22

Hints

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Step-by-step solution

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1. Figure: a circle with centre O and diameter AB. Tangent XY touches the circle at A and tangent PQ touches it at B, with X, P on the same side of AB. Given: XY and PQ are tangents at the ends of diameter AB. To prove: $XY\parallel PQ$.

   ½ mark

2. OA is a radius and XY is the tangent at A, so $OA\perp XY$ [tangent ⟂ radius at the point of contact]. Hence $\angle OAY=90^{\circ }$, i.e. $\angle BAY=90^{\circ }$.

   ½ mark

3. OB is a radius and PQ is the tangent at B, so $OB\perp PQ$ [tangent ⟂ radius]. Hence $\angle OBP=90^{\circ }$, i.e. $\angle ABP=90^{\circ }$.

   ½ mark

4. AB is a transversal of XY and PQ, and $\angle BAY$ and $\angle ABP$ are alternate interior angles. They are equal (each $90^{\circ }$), so $XY\parallel PQ$.

   ½ mark

Answer: XY ∥ PQ, because the alternate angles ∠OAY and ∠OBP are both 90°.

Revise the chapter

• Chapter: Circles (notes, formulas, where marks are lost and every past question)
• Topic: Solving Problems Using Tangent Properties