Exercise 8A

A point $P$ is at a distance of $29 \mathrm{cm}$ from the centre of a circle of radius $20 \mathrm{cm}$. Find the length of the tangent drawn from $P$ to the circle.

A point $P$ is $25 \mathrm{cm}$ away from the center of a circle and the length of tangent drawn from $P$ to the circle is $25 \mathrm{cm}$. Find the radius of the circle. If the required answer is $x \mathrm{cm}$, then what is the value of $x$?

In the given figure, the chord $\mathrm{AB}$of the larger of the two concentric circles, with center $\mathrm{O}$, touches the smaller circle at $\mathrm{C}$. Prove that $\mathrm{AC}=\mathrm{CB}$.

From an external point $\mathrm{P}$, tangents $\mathrm{PA}$ and $\mathrm{PB}$ are drawn to a circle with center $\mathrm{O}$. If $\mathrm{CD}$ is the tangent to the circle at a point $\mathrm{E}$ and $\mathrm{PA}=14 \mathrm{cm}$, find the perimeter of $\mathrm{\Delta PCD}$.

In the given figure, a circle touches all the four sides of a quadrilateral $\mathrm{ABCD}$ whose three sides are $\mathrm{AB}=6 \mathrm{cm}, \mathrm{BC}=7\mathrm{cm}$ and $\mathrm{CD}=4 \mathrm{cm}$. Find $\mathrm{AD}$.

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In the given figure, a triangle $\mathrm{ABC}$ is drawn to circumscribe a circle of radius $3 \mathrm{cm}$ such that the segments $\mathrm{BD}$and $\mathrm{DC}$into which $\mathrm{BD}$ is divided by the point of contact $\mathrm{D}$, are of lengths $6 \mathrm{cm}$and $9 \mathrm{cm}$ respectively. If the area of $\mathrm{\Delta ABC} = 54 {\mathrm{cm}}^2$ then find the lengths of sides $\mathrm{AB}$and $\mathrm{AC}$.

$\mathrm{PQ}$is a chord of length $4.8 \mathrm{cm}$ of a circle of radius $3 \mathrm{cm}$. The tangents at $\mathrm{P}$and $\mathrm{Q}$intersect at a point $\mathrm{T}$as shown in the figure. Find the length of $\mathrm{TP}$.

In the given figure, $\mathrm{O}$ is the centre of the circle and $\mathrm{TP}$ is the tangent to the circle from an external point $\mathrm{T}$. If $\angle \mathrm{PBT} = 30°$ , prove that $\mathrm{BA}:\mathrm{AT}=2:1$.