In the following figures, $O$ is the centre of circles. Find the values of $x$ and $y$.
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In $\odot O$, chords $A B$ is perpendicular to $C D$ at $P, A B=16, C P=4 P D=10 .$ Find the radius.
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In the figure, $O$ is the centre of the concentric circles and $O N \perp A B .$ If $O C=10, O N=8$ and $O B=17$ find $A C$.
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Prove Theorem $10$: Of any two chords of a circle, the greater chord is nearer to the centre, and conversely, the chord nearer to the centre is larger.
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Let $P$ be a point inside a circle. $A B$ is the diameter passes through $P$ and $C P D$ is the chord perpendicular to $A B .$ Show that $C D$ is the shortest of all chords passing through $P$.
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Through a point $P$ in a circle, the longest chord that can be drawn is $10$ cm long and the shortest chord is $6$ cm long. What is the radius of the circle and how far is $P$ from the centre?
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In $\odot O$, chords $A B$ and $C D$ are equal and intersect in the circle at $E$ such that $A E<E B$ and $C E<E D .$ Show that $\triangle B D E$ is isosceles with base $B D$
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In $\odot O$, congruent chords $A B$ and $C D$ are produced to meet at $P$. Prove that $\triangle P A C$ is isosceles.
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In $\odot O, A B$ and $B C$ are equal chords, $O V \perp A B$, and $O U \perp B C$. Prove that. $B$ is the midpoint of $\operatorname{arc} V U$.
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In parallelogram $P Q R S$, $P Q=5$ cm $P R=8$ cm, $Q S=6$ cm. Calculate the lengths of $A R$ and $B R$.
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Chords $A B$ and $C D$ intersect at $E$ and $A E=E B$. A semicircle is drawn with diameter $C D . E F$, perpendicular to $C D$, meets this semicircle at $F$. Prove that $A E=E F$.
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$\odot O$ and $\odot P$ intersect at $A$ and $B .$ Show that $O P$ is the perpendicular bisector of the common chord $A B$.
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