Exercise 15.3

Two chords $AB\mathit{,}\mathit{ }CD$ of a circle intersect internally at a point $P$. If $AP=5 \mathrm{cm}$, $PB=6 \mathrm{cm}$$\text{ and CD}=13 \mathrm{cm}$, $\text{ find CP}.$

In the figure given below, diameter $\mathrm{AB}$and chord $\mathrm{CD}$ of a circle meet at $\mathrm{P}. \mathrm{PT}$ is a tangent to the circle at $\mathrm{T}$. $\mathrm{CD}=7.8 \mathrm{cm}, \mathrm{PD}=5 \mathrm{cm}, \mathrm{PB}=4 \mathrm{cm}$. Find: $\mathrm{AB}$

In the figure given below, diameter $\mathrm{AB}$and chord $\mathrm{CD}$ of a circle meet at $\mathrm{P}. \mathrm{PT}$ is a tangent to the circle at $\mathrm{T}$. $\mathrm{CD}=7.8 \mathrm{cm}, \mathrm{PD}=5 \mathrm{cm}, \mathrm{PB}=4 \mathrm{cm}$. Find the length of tangent $\mathrm{PT}$.

In the adjoining figure, $CBA$ is a secant and $CD$ is tangent to the circle. If $AB=7 \mathrm{cm}$ and $BC=9 \mathrm{cm}$, then, Prove that $△ACD~△DCB$.

In the adjoining figure, $CBA$ is a secant and $CD$ is tangent to the circle. If $AB=7 \mathrm{cm}$ and $BC=9 \mathrm{cm}$, then find the length of $CD$.

In the figure given below, two circles intersect at $A,B$. From a point $P$ on one of these circles, two line segments $PAC$ and $PBD$ are drawn, intersecting the other circle at $C$ and $D$ respectively. Prove that $CD$ is parallel to the tangent at $P$.

In the figure given below, $O$ and $O\text{'}$ are centres of two circles touching each other externally at the point $P$. The common tangent at $P$ meets a direct common tangent $AB$ at $M$. Prove that: $M$ bisects $AB$ 

In the figure given below, $O$ and $O\text{'}$ are centres of two circles touching each other externally at the point $P$. The common tangent at $P$ meets a direct common tangent $AB$ at $M$. Prove that: $\angle APB=90°$