Circles

In the given figure, $XY$ is a tangent to the circle with center $O$ at the point of contact $P$. If $Q$ is any point on the tangent $XY$ other than $P$, and $OQ$ intersects the circle at point $R$, which of the following statements is true regarding the relationship between lengths $OP$ and $OQ$?

The line that intersects a circle at two points is called a:

If the radius of a circle is $15$ cm and the distance of a point $P$ from the center is $17$ cm, find the length of the tangent from $P$ to the circle.

A tangent $L$ is drawn to a circle with center $O$ at point $M$. If $N$ is any other point on the tangent $L$, then:

In the figure, $AB$ is a chord of length $8$ cm of a circle of radius $5$ cm. The tangents at $A$ and $B$ intersect at point $P$. Find the length $AP$.

If a line $y = 3$ is tangent to a circle with center $(0,0)$, what is the radius of the circle?

Two tangents $TP$ and $TQ$ are drawn to a circle with center $O$ from an external point $T$. If $\angle PTQ = 70^{\circ}$, then $\angle OPQ$ is:

A surveyor needs to find the distance between a point $A$ on a circular track and a point $B$ outside. The line $AB$ is tangent to the track at $A$. If the radius of the track is $40$ m and the distance from the center $O$ to $B$ is $50$ m, what is the length $AB$?

In a circle with center $O$, $PQ$ is a tangent at point $P$. A point $R$ is taken on the circle such that $PR$ is a chord. If \\angle QPR = 50^\\circ, find the measure of $\\angle POR$ using the fact that $OP \\perp PQ$.

A square $ABCD$ is circumscribed about a circle. If the radius of the circle is $5$ cm, what is the length of the diagonal of the square? (Hint: The tangent at each point of contact is perpendicular to the radius).

The length of the tangent drawn from a point $8$ cm away from the center of a circle of radius $6$ cm is:

How many tangents can be drawn to a circle from a point outside it?

The distance between two parallel tangents of a circle of radius $4$ cm is:

Two tangents $PA$ and $PB$ are drawn to a circle with center $O$ from an external point $P$. Which of the following is true?

In the given figure, $PA$ and $PB$ are tangents to a circle with center $O$ from an external point $P$. If $\angle APB = 60^\circ$ and the radius of the circle is $5$ cm, what is the length of the tangent $PA$?

In the figure, if $AD, AE$ and $BC$ are tangents to the circle, and $AD = 10$ cm, then the perimeter of $\triangle ABC$ is:

In the figure, a circle is inscribed in a $\triangle ABC$ having sides $8$ cm, $10$ cm and $12$ cm. Find the length of $AD$ where $D$ is the point of contact on $AB$ and the circle touches $AB, BC, CA$ at $D, E, F$ respectively.

If two tangents $PA$ and $PB$ are drawn to a circle with center $O$ from an external point $P$ such that $\angle AOB = 110^\circ$, find the measure of $\angle OPA$.

A triangle $ABC$ is drawn to circumscribe a circle of radius $3$ cm such that the segments $BD$ and $DC$ into which $BC$ is divided by the point of contact $D$ are of lengths $6$ cm and $9$ cm respectively. Find the length of side $AB$.

A point $P$ is $13$ cm from the center of a circle. The length of the tangent drawn from $P$ to the circle is $12$ cm. If another point $Q$ is at a distance of $10$ cm from the center, what is the length of the tangent from $Q$ to the same circle?

The number of tangents that can be drawn from a point inside the circle is:

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

If $\angle POQ = 130^{\circ}$, where $OP$ and $OQ$ are radii of a circle with center $O$, then the angle between the tangents at $P$ and $Q$ is:

A point $P$ is $26$ cm away from the center of a circle and the length of the tangent $PT$ drawn from $P$ to the circle is $24$ cm. Find the radius of the circle.

Two tangents $PA$ and $PB$ are drawn to a circle with center $O$ from an external point $P$. If $\angle OBA = 10^{\circ}$, then $\angle APB$ is:

In the given figure, $PA$ and $PB$ are tangents to the circle from an external point $P$. $CD$ is another tangent touching the circle at $Q$. If $PA = 12$ cm, find the perimeter of $\triangle PCD$.

In the given figure, $PQ$ is a chord of length $8$ cm of a circle of radius $5$ cm. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length of $TP$.

From a point $P$, two tangents $PA$ and $PB$ are drawn to a circle with center $O$. If $OP$ is equal to the diameter of the circle, find the measure of $\angle APB$.

In a right triangle $ABC$, right-angled at $B$, $BC = 6$ cm and $AB = 8$ cm. A circle is inscribed in the triangle. Find the radius of the circle.

The radii of two concentric circles are $13$ cm and $5$ cm. A chord of the larger circle is tangent to the smaller circle. Find the length of this chord.

Which of the following statements is true for a circle?

A point $P$ is $26\text{ cm}$ away from the centre of a circle and the length of the tangent drawn from $P$ to the circle is $24\text{ cm}$. Find the radius of the circle.

The distance between two parallel tangents of a circle of radius $4\text{ cm}$ is:

The length of a tangent from a point $A$ at distance $5\text{ cm}$ from the centre of the circle is $4\text{ cm}$. The radius of the circle is:

In the following figure, $PA$ and $PB$ are tangents to the circle with centre $O$ from an external point $P$. If $\angle APB = 60^\circ$, find the measure of $\angle OAB$.

In the given figure, $PT$ is a tangent to a circle with centre $O$ from an external point $P$. If $OP = 13$ cm and $OT = 5$ cm, what is the length of the tangent segment $PT$?

In two concentric circles, a chord of length $24\text{ cm}$ of the larger circle touches the smaller circle. If the radius of the larger circle is $13\text{ cm}$, then the radius of the smaller circle is:

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

The tangents at the end points of a diameter of a circle are parallel. If the distance between two such parallel tangents is $14$ cm, and a point $P$ is $25$ cm from the center, what is the length of the tangent from $P$ to the circle?

Two parallel tangents to a circle with center $O$ are intercepted by a third tangent at points $X$ and $Y$. What is the measure of the angle $\angle XOY$?

The distance between two parallel tangents of a circle of radius $4$ cm is:

The lengths of tangents drawn from an external point to a circle are:

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

If $\angle POQ = 110^{\circ}$ in a circle with center $O$ and tangents $TP$, $TQ$, find $\angle PTQ$.

Two concentric circles have radii $13$ cm and $5$ cm. A chord $AB$ of the larger circle touches the smaller circle at point $P$. Calculate the length of the chord $AB$.

In the given figure, $PA$ and $PB$ are tangents to a circle with center $O$ from an external point $P$. If $\angle APB = 60^\circ$ and the radius of the circle is $5$ cm, find the length of the chord $AB$.

If two tangents inclined at an angle of $60^{\circ}$ are drawn to a circle of radius $3$ cm, then the length of each tangent is:

At the point of contact of two circles, a common tangent is drawn. If the radii of the circles are $9$ cm and $4$ cm and they touch each other externally, find the length of the direct common tangent.

In the following figure, a circle touches all four sides of a quadrilateral $ABCD$. If $AB = 6$ cm, $BC = 7$ cm, and $CD = 4$ cm, find the length of $AD$.

A chord $PQ$ of a circle of radius $5$ cm is of length $8$ cm. The tangents at $P$ and $Q$ intersect at a point $T$. Find the length of $TP$.

In the given figure, a point $P$ lies exactly on the circumference of a circle with center $O$. How many tangent(s) can be drawn to the circle passing through this point $P$?

The distance between two parallel tangents of a circle of radius $4$ cm is:

A tangent $PQ$ at a point $P$ of a circle of radius $5$ cm meets a line through the center $O$ at a point $Q$ so that $OQ = 12$ cm. What is the length $PQ$?

How many tangents can be parallel to a secant of a circle?

In the figure, $O$ is the center of the circle, $AB$ is a chord and $AT$ is the tangent at $A$. If $\angle BAT = 60^{\circ}$, then $\angle AOB$ is:

From a point $P$ which is at a distance of $13$ cm from the center $O$ of a circle of radius $5$ cm, a pair of tangents $PQ$ and $PR$ to the circle are drawn. What is the area of the quadrilateral $PQOR$?

Two tangents $PA$ and $PB$ are drawn to a circle with center $O$ from an external point $P$. If $\angle OBA = 10^{\circ}$, then $\angle APB$ is:

A landscape architect is designing a circular flower bed with a path represented by a point $P$ located exactly on the boundary of the flower bed. A straight water pipe needs to be laid such that it is tangent to the circular bed at point $P$. Based on the geometric properties of circles, how many such distinct tangent lines can be drawn to the circle that pass through this specific point $P$?

A point $P$ is at a distance of $13$ cm from the center $O$ of a circle of radius $5$ cm. Two tangents $PQ$ and $PR$ are drawn to the circle. What is the area of the quadrilateral $PQOR$?

If two tangents inclined at an angle of $60^{\circ}$ are drawn to a circle of radius $3$ cm, then the length of each tangent is: