Q. In the adjoining figure $$BC$$ is a diameter of a circle with centre $$O$$. If $$AB$$ and $$CD$$ are two chords such that $$AB\parallel CD$$, prove that $$AB=CD$$

Q. $$AB$$ and $$AC$$ are two chords of a circle of radius $$r$$ such that $$AB=2AC$$. If $$p$$ and $$q$$ are the distances of $$AB$$ and $$AC$$ from the centre then prove that $$4q^{2}=p^{2}+3r^{2}$$

Q. In the adjoining figure, two circles with centres $$A$$ and $$B$$, and of radii $$5\ cm$$ and $$3\ cm$$ touch each other internally. If the perpendicular bisector of $$AB$$ meets the bigger circle in $$P$$ and $$Q$$, find the length of $$PQ$$.

Q. In a circle of radius $$5\ cm, AB$$ and $$CD$$ are two parallel chords of lengths $$8\ cm$$ and $$6\ cm$$ respectively.
Calculate the distance between the chords if they are on opposite sides of the centre.

Q. In a circle of radius $$5\ cm, AB$$ and $$CD$$ are two parallel chords of lengths $$8\ cm$$ and $$6\ cm$$ respectively.
Calculate the distance between the chords if they are on the same side of the centre

Q. A chord of length $$30\ cm$$ is drawn at a distance of $$8\ cm$$ from the centre of a circle. Find out the radius of the circle.

Q. Find the length of a chord which is at a distance of $$3\ cm$$ from the centre of a circle of radius $$5\ cm$$

Q. Prove that the diameter of a circle perpendicular to one of the two parallel chords of a circle is perpendicular to the other and bisects it.

Q. In the adjoining figure, $$O$$ is the centre of a circle. If $$AB$$ and $$AC$$ are chords of the circle such that $$AB=AC, OP\bot AB$$ and $$OQ\bot AC$$, prove that $$PB=QC$$

Q. In the given figure, the diameter $$CD$$ of a circle with centre $$O$$ is perpendicular to chord $$AB$$. If $$AB=12\ cm$$ and $$CE=3\ cm$$ calculate the radius of the circle.

Q. A chord of length $$16\ cm$$ is drawn in a circle of radius $$10\ cm$$. Find the distance of the chord from the centre of the circle.

Q. In the given figure, $$O$$ is the centre of a circle in which chords $$AB$$ and $$CD$$ intersect at $$P$$ such that $$PO$$ bisect $$\angle BPD$$. Prove that $$AB=CD$$.

Q. If a diameter of a circle bisects each of the two chords of a circle then prove that the chords are parallel.

Q. Prove that two different circles cannot intersect each other at more than two points.

Q. In the adjoining figure $$OD$$ is perpendicular to the chord $$AB$$ of a circle with centre $$O$$. If $$BC$$ is a diameter, show that $$AC\parallel DO$$ and $$AC=2\times OD$$

Q. Two parallel chords of lengths $30cm$ and $16cm$ are drawn on the opposite sides of the centre of a circle of radius $17cm$. Find the distance between the chords.

Q. In the given figure, a circle with centre $O$ is given in which a diameter $AB$ bisect the chord $CD$ at a point $E$ such that $CE=ED=8cm$ and $EB=4cm$. Find the radius of the circle.

Q. Two circles of radii $10cm$ and $8cm$ intersect each other, and the length of the common chord is $12cm$. Find the distance between their centres.

Q. Two equal circles intersect in $P$ and $Q$. A straight line through $P$ meets the circles in $A$ and $B$. prove that $QA=QB$

Q. In the given figure, $AB$ is a chord of a circle with center $O$ and $AB$ is produced to $C$ such that $BC=OB$. Also, $CO$ is joined and produced to meet the circle in $D$. If $\angle ACD=y^o$ and $\angle AOD=x^o$, prove that $x=3y$