Tangent to a Circle

If$AP=12 cm, OP=13 cm$, then $CD=?$

In the given figure, find $\mathrm{TP}$ if $\mathrm{AT}=16\mathrm{cm}$ and $\mathrm{AB}=12\mathrm{cm}.$ (in $\mathrm{cm}$)

In the following figure, $\mathrm{PQ}$ and $\mathrm{PR}$ are tangents to the circle, with centre $\mathrm{O}.$ If $\angle \mathrm{QPR} = 60°,$ calculate $\angle \mathrm{QOR}.$

In a triangle $\mathrm{ABC},$ the incircle with centre $\mathrm{O}$ touches $\mathrm{BC}, \mathrm{CA}$and $\mathrm{AB}$ at points $\mathrm{P},$ $\mathrm{Q}$ and $\mathrm{R}$ respectively. Calculate $\angle \mathrm{QOR}$, given that $\angle \mathrm{A} = 60°.$

In the following figure, $\mathrm{PQ}$ is the tangent to the circle at $\mathrm{A}, \mathrm{DB}$ is the diameter and $\mathrm{O}$ is the centre of the circle. If $\angle \mathrm{ADB} = 30°$ and $\angle \mathrm{CBD} = 60°,$ calculate $\angle \mathrm{PAD}.$

The given figure shows a circle with centre $\mathrm{O}$ and $\mathrm{BCD}$ is tangent to it at $\mathrm{C}.$ Show that $\angle \mathrm{ACD}+\angle \mathrm{BAC}=90°.$

In the given figure, $\mathrm{MN}$ is the common chord of two intersecting circles and $\mathrm{AB}$ is their common tangent. Prove that the line $\mathrm{NM}$ produced bisects $\mathrm{AB}$ at $\mathrm{P}.$

In two concentric circles, prove that all chords of the outer circle, which touch the inner circle, are of equal length.

Two parallel tangents of a circle meet a third tangent at points $\mathrm{P}$ and $\mathrm{Q}.$ Prove that $\mathrm{PQ}$ subtends a right angle at the centre.
 

Tangents $\mathrm{AP}$ and $\mathrm{AQ}$ are drawn to a circle, with centre $\mathrm{O},$ from an exterior point $\mathrm{A}.$ Prove that $\angle \mathrm{PAQ} = 2\angle \mathrm{OPQ}$.
 

In figure, $O$ is the centre of the circle and from an exterior point $\text{'}P\text{'}$ two tangents$PA$ and $PB$ are drawn to the circle at$A$and $B$respectively then prove that $PAOB$ is a cyclic quadrilateral. 
 

A circle touches all sides of a quadrilateral. Prove that the angle made by opposite sides at the centre are supplementary.

A circle with centre at $\text{'}O\text{'}$ touches the four sides of a quadrilateral $ABCD$ internally in such a way that it divides $AB$ in $3:1$ and $AB= 8 \mathrm{cm}$. If the radius of the circle is $x \mathrm{cm}$, where $OA = 10 \mathrm{cm}$, find the value of $x$.

The tangent of length $4 \mathrm{cm}$ is drawn to a circle, which is $10 \mathrm{cm}$ away from the centre of the circle. If the radius of the circle is $k\sqrt{21} \mathrm{cm}$. Find the value of $k$.

The common point of tangent line and circle is known as point of _____ . 

A circle can have _____ parallel tangent lines.

_____ tangent line(s) can be drawn from a point situated on circle. (Mention the answer in words)

$"LM and XY"$ two tangents at the points $"P"$ and $"Q"$on the circle if $"PQ"$is diameter then $LM\parallel XY$.

A tangent line $XY,$touches a circle at point $P$ whose center is $O$ and $Q$ is any other point on the tangent line, then $OP=OQ$ .

Tangent of circle is the line which intersect the circle at two points.