Q. Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Q. Prove that the parallelogram circumscribing a circle is a rhombus.

Q. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Q. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Q. From a point $Q$, the length of the tangent to a circle is $24$ cm and the distance of $Q$ from the centre is $25$ cm. The radius of the circle is-

1. $7$ cm
2. $12$ cm
3. $15$ cm
4. $24.5$ cm

Q.

Q. Why did it make Mr Purcell feel “vaguely insulted”?

Q. The length of a tangent from a point $A$ at distance $5$ cm from the centre of the circle is $4$ cm. Find the radius of the circle.

Q. If tangents $PA$ and $PB$ from a point $P$ to a circle with centre $O$ are inclined to each other at angle of ${80}^o$, then $\angle POA$ is equal to.

1. ${50}^o$
2. ${60}^o$
3. ${80}^o$
4. ${70}^o$

Q. In Figure, $XY$ and $X^{\prime }{Y}^{\prime }$ are two parallel tangents to a circle with centre $O$ and another tangent $AB$ with point of contact $C$ intersecting $XY$ at $A$ and $X^{\prime }{Y}^{\prime }$ at $B$. Prove that $\angle$$AOB$$={90}^o$.

Q. How many tangents can a circle have?

Q. A triangle $ABC$ is drawn to circumscribe a circle of radius $4cm$ such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of the lengths $8cm$ and $6cm$ respectively. Find the sides $AB$ and $AC$.

Q. A tangent $PQ$ at a point $P$ of a circle of radius $5$cm meets a line through the centre $O$ at a point $Q$ so that OQ$=12$ cm. Length $PQ$ is:

1. $12$ cm
2. $8.5$ cm
3. $13$ cm
4. $\sqrt{119}$ cm

Q. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Q. Two concentric circles are of radii $5$ cm and $3$ cm. Find the length of the chord of the larger circle which touches the smaller circle.

Q. Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.

Q. In Figure If $TP$ and $TQ$ are the two tangents to a circle with centre $O$ so that $\angle POQ$ $={110}^o$, the $\angle$ $PTQ$ is equal to.

1. ${70}^o$
2. ${90}^o$
3. ${80}^o$
4. ${60}^o$