Execise 8

In the adjoining figure, $PA$ and $PB$ are tangents to the circle from an external point $P$. $CD$ is another tangent touching the circle at $Q$. If $AP=12 \mathrm{cm}$ and $CQ=DQ=3 \mathrm{cm}$, then find $CP+DP$.

Let $s$ denotes the semi-perimeter of $△ABC$ in which $\mathrm{BC}=\mathrm{a}, \mathrm{CA}=\mathrm{b}, \mathrm{AB}=\mathrm{c}$. If a circle touches the sides $\mathrm{BC}, \mathrm{CA}, \mathrm{AB}$ at $\mathrm{D}, \mathrm{E}, \mathrm{F}$ respectively. Prove that $\mathrm{BD}=\mathrm{s}-\mathrm{b}$.

In the adjoining figure, quadrilateral $ABCD$ is circumscribed. If the radius of incircle (centre $O$) is $10 \mathrm{cm}$ and $AD\perp DC$. Find the value of $x$.

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

In the adjoining figure, $ABC$ is a triangle in which $\angle B=90°$, $BC=48 \mathrm{cm}$ and $AB=14 \mathrm{cm}$. A circle is inscribed in the triangle whose centre is $O$. Find the radius of the incircle.

In the adjoining figure, $AB$ is a chord of a circle with centre $O$. At $B,$a tangent $PB$ is drawn such that its length is $24 \mathrm{cm}$. The distance of $P$ from centre is $26 \mathrm{cm}$. If the length of chord $AB$ is $16 \mathrm{cm}$, then find its distance from the centre.

In the adjoining figure, a triangle $PQR$ is drawn to circumscribe a circle of radius $6 \mathrm{cm}$ such that the segments $QT$ and $QR$ into which $QR$ is divided by the point of contact $T$, are of lengths $12 \mathrm{cm}$ and $9 \mathrm{cm}$ respectively. If the area of $△PQR$ is $189 {\mathrm{cm}}^2$, then find the lengths of $PQ$ and $PR$.

In the adjoining figure, $AB$ is a chord of length $16 \mathrm{cm}$ of a circle with centre $O$ and of radius $10 \mathrm{cm}$. The tangents at $A$ and $B$ intersect at the point $P$. Find the length of $PA$.