Question

If a line intersect two concentric circles with centre A in points P,Q, R and S respectively then prove that PQ = RS. (Fig.4.35)

Answer 1

Draw a perpendicular from centre A on the line which passes through P, Q, R and S.
Now, considering the smaller concentric circle,
A is the centre, QM is chord and AM$\perp \mathrm{QR}$.
Therefore, QM = MR -----------(1)
(perpendicular from the centre bisects the respective chord)
And, considering the larger concentric circle,
A is the centre, PS is the chord and AM$\perp \mathrm{PS}$.
Therefore, PM = MS -----------(2)
(perpendicular from the centre bisects the respective chord)
Subtracting equation (1) from equation (2), we get:
PM $-$ QM = MS $-$ MR
Therefore, PQ = RS.
Hence proved.