Chapter: Circle· Mathematics Part 2 (Geometry)

In figure 3.101, two circles intersect at points M and N. Secants drawn through M and N intersect the circles at points R, S and P, Q respectively. Prove that seg SQ || seg RP.

In figure 3.101, two circles intersect at points M and N. Secants drawn through M and N intersect the circles at points R, S and P, Q respectively. Prove that : seg SQ || seg RP.

[Figure: A smaller circle on the left and a bigger circle on the right intersect at M (upper) and N (lower). Secant R–M–S: R is on the smaller circle and S on the bigger circle. Secant P–N–Q: P is on the smaller circle and Q on the bigger circle. Segments RP and SQ are drawn.]

Doubtly Asked 4 Nov 2026 Student doubt Save WhatsApp Report

1 answer

Construction: Draw seg MN.

Proof:

\(\square RMNP\) is cyclic (its vertices lie on the smaller circle) and P – N – Q.

\(\therefore \angle MRP = \angle MNQ\) ...(1) (exterior angle of a cyclic quadrilateral is equal to the angle opposite to its adjacent interior angle)

\(\square MNQS\) is cyclic (its vertices lie on the bigger circle).

\(\therefore \angle MNQ + \angle MSQ = 180^\circ\) ...(2) (opposite angles of a cyclic quadrilateral are supplementary)

From (1) and (2), \(\angle MRP + \angle MSQ = 180^\circ\)

i.e. \(\angle PRS + \angle RSQ = 180^\circ\) (R – M – S)

These are interior angles on the same side of transversal RS for lines RP and SQ, and they are supplementary.

\(\therefore\) seg SQ || seg RP (interior angles test)

Doubtly AI · 4 Nov 2026 AI-verified · Nov 2026 Report

Was this answer helpful?

In figure 3.101, two circles intersect at points M and N. Secants drawn through M and N intersect the circles at points R, S and P, Q respectively. Prove that seg SQ || seg RP.

1 answer

Your answer