Circles
Circle
nta_abhyas_2025
Grade 11
Question:
Let $PQ$ and $RS$ be the tangents at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle, then the value of $2r$ is equal to
\sqrt{PQ \cdot RS}
\frac{PQ + RS}{2}
\frac{PQ \cdot RS}{PQ + RS}
\sqrt{\frac{PQ^2 + RS^2}{2}}
Step-by-Step Solution
Key Concept: When two chords intersect inside a circle, similar triangles are formed whose corresponding sides establish relationships between the chord lengths.
From the figure, it is clear that triangles $\triangle PQO$ and $\triangle RSO$ are similar. This similarity implies that corresponding sides are proportional. By the properties of similar triangles and the symmetric configuration shown, we have $\frac{PQ}{RS} = \frac{PO}{RO}$. However, since the triangles are congruent (not just similar), we get $PQ = RS$ directly. Additionally, $PK = \sqrt{PQ \cdot RS}$ confirms this relationship.
Correct Answer: 1