Short Answer Questions
1. Out of the two concentric circle ,the radius of the outer circle is 5cm and the chord FC is of length 8cm is a tangent to the inner circle .Find the radius of the inner circle.
Sol. Let the chord FC of the larger circle touch the smaller circle at the point L.
Since FC is tangent at the point L to the smaller circle with the centre O.
Since AC is chord of the bigger circle and OL
∴ OL bisects FC
Now, consider right-angled
Hence, the radius of the smaller or inner circle is 3cm
2. Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.
Join XB and XF
XB=XF (radii of same circle)
(tangents from an external point)
(by SSS congruence rule)
3.In the given figure,LM and NP are common tangents to two circles of unequal radii.Prove that LM=NP.
Sol. Let Chords LM and NP meet at the point R
Since RL and RN are tangents from an external point R to two Circles with centres O and
∴ RL=RN …..(i)
And RM=RP …..(ii)
Subtracting (ii) from (i),we have
4.In the given figure, common tangents PQ and RS to two circles intersect at T. Prove that PQ=RS.
Sol. Clearly, TP and TR are two tangents from an external point T to the circle with centre O.
∴ TP=TR ……(i)
Also, TQ and TS are two tangents from an external point T to the circle with centre
∴ TQ=TS …..(ii)
Adding (i) and (ii),we obtain
5.A chord XY of a circle is parallel to the tangent drawn at a point Z of the circle. Prove that Z bisects the arc XZY.
Sol. Since XY is parallel to the tangent drawn at the point Z and radius OZ is perpendicular to the tangent.
∴ OL bisects the chord XY.
∴ arc XZ=arc ZY
i.e., Z bisects arc XZY