REVISION EXERCISE

In the following figure, if $\mathrm{PT}$ is a tangent of the circle with centre $O$ and $\angle \mathrm{TPO}=25°$, then the measure of $x$ is:

Two concentric circles are of radii $10 \mathrm{cm}$ and $8 \mathrm{cm}$. $\mathrm{RP}$ and $\mathrm{RQ}$ are tangents to the two circles from $\mathrm{R}$. If the length of $\mathrm{RP}$ is $24 \mathrm{cm}$, If the length of $\mathrm{RQ}=k\sqrt{17} cm$, then find the value of $k$.

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If from an external point $\mathrm{B}$ of a circle with centre $\mathrm{O}$, two tangents $\mathrm{BC}$ and $\mathrm{BD}$ are drawn such that $\angle \mathrm{DBC}=120°$. Prove that $\mathrm{BO}=2\mathrm{BC}$.

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

In figure, $\mathrm{OP}$ is equal to the diameter of the circle. Prove that $\mathrm{ABP}$ is an equilateral triangle.

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Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

In figure, $\mathrm{PO}\perp \mathrm{QO}$, the tangents to the circle with centre $\mathrm{O}$ at $\mathrm{P}$ and $\mathrm{Q}$ intersect at point $\mathrm{T}$. Prove that $\mathrm{PQ} \mathrm{and} \mathrm{OT}$ are right bisectors of each other.

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In figure, from an external point $\mathrm{P}$, two tangents, $\mathrm{PA}$ and $\mathrm{PB}$ are tangents to the circle with centre $\mathrm{O}$. If $\mathrm{CD}$ is another tangent at point $\mathrm{E}$ to the circle and $\mathrm{PA}=12 \mathrm{cm}$. Find the perimeter of $∆\mathrm{PCD}$.