Shown below is a circle and$2$ congruent squares $(\mathrm{PQRS} \& \mathrm{QTUR})$. $\mathrm{ST}, \mathrm{SU}$ and $\mathrm{UT}$are tangents to the circle. The side length of the square is $10 \mathrm{cm}$.

Find the radius of the circle. Show your work.

In the given figure, $\mathrm{AB}$ is a diameter of the circle with centre $\mathrm{O}. \mathrm{AP}, \mathrm{BQ} \mathrm{and} \mathrm{PRQ}$ are tangents. Prove that $\angle \mathrm{POQ} = 90°$.

In the given figure,$\mathrm{PQ}, \mathrm{PR}$ and $\mathrm{ST}$ are tangents to the same circle. If $\angle \mathrm{P} = 40°$ and $\angle \mathrm{QRT} = 75°$, find $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}.$

In a right triangle $ABC$, a circle with a side $AB$ as diameter is drawn to intersect the hypotenuse $AC$ in $P$. Prove that the tangent to the circle at $P$ bisects the side $BC$.

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In the figure below, O is the centre of two concentric circles. $∆\mathrm{PQR}$ is an equilateral triangle such that its vertices and sides touch the bigger and smaller circles respectively. The difference between the area of the bigger circle and the smaller circle is 616 cm².

Find the perimeter of $∆\mathrm{PQR}.$

Shown below is a circle with centre M. PQ is a secant and $\angle \mathrm{KML} = \mathrm{\theta }$

i) Show that, when $\mathrm{\theta }=0°$ PQ becomes a tangent to the circle.

ii) What is the point of contact of the tangent in part i) with the circle?

Shown below is a circle whose centre is unknown.

State true or false for the statements below and give valid reasons.

i) The centre of the circle can be found using any 2 tangents.
ii) The centre of the circle can be found using any 2 chords.