Optional Exercise

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

$\mathrm{PQ}$ is a chord of length $8 \mathrm{cm}$ of a circle of radius $5 \mathrm{cm}$. The tangents at $\mathrm{P}$ and $\mathrm{Q}$ intersect at a point $\mathrm{T}$ (See figure). Find the length of $\mathrm{TP}$.

Prove that opposite sides of a quadrilateral circumscribing a circle subtends supplementary angles at the centre of the circle.

Draw a line segment $\mathrm{AB}$ of length $8 \mathrm{cm}$. Taking $\mathrm{A}$ as centre, draw a circle of radius $4 \mathrm{cm}$ and taking $\mathrm{B}$ as centre, draw another circle of radius $3 \mathrm{cm}$. Construct tangents to each circle from the centre of the other circle.

Let $\mathrm{ABC}$ be a right triangle in which $\mathrm{AB}=6 \mathrm{cm}, \mathrm{BC}=8 \mathrm{cm}$ and $\angle \mathrm{B}=90°$. $\mathrm{BD}$ is the perpendicular from $\mathrm{B}$ on $\mathrm{AC}$. The circle through $\mathrm{B}, \mathrm{C}, \mathrm{D}$ is drawn. Construct the tangents from $\mathrm{A}$ to this circle.

Find the area of the shaded region (in ${\mathrm{cm}}^2$) in the figure, in which two circles with centres $\mathrm{A}$ and $\mathrm{B}$ touch each other at the point $\mathrm{C}$, where $\mathrm{AC}=8 \mathrm{cm}$ and $\mathrm{AB}=3 \mathrm{cm}$.

$ABCD$ is a rectangle with $AB=14 \mathrm{cm}$ and $BC=7 \mathrm{cm}$. Taking $DC, BC$ and $AD$ as diameters, three semicircles are drawn as shown in the figure. Find the area of the shaded region in ${\mathrm{cm}}^2$.