Exercise

In the figure below, $∆\mathrm{PXY}$ is formed using three tangents to a circle centred at $\mathrm{O}$.

(Note: The figure is not to scale)

Based on the construction, the sum of the tangents $\mathrm{PA}$ and $\mathrm{PB}$ is_____the perimeter of $∆\mathrm{PXY}$.

Raghav drew the following figure on a board where a circle is inscribed in a quadrilateral.

(Note: The figure is not to scale)

Then he wrote the following relationships.

i) $\mathrm{ZW}+\mathrm{WX}=\mathrm{XY}+\mathrm{YZ}$

ii) $\mathrm{ZY}+\mathrm{WX}=\mathrm{ZW}+\mathrm{YX}$

Which of the above relationships is/are DEFINITELY true?

A circle has a centre $\mathrm{O}$ and radii $\mathrm{OQ}$ and $\mathrm{OR}$. Two tangents, $\mathrm{PQ}$ and $\mathrm{PR}$, are drawn from an external point $\mathrm{P}$.

In addition to the above information, which of these must also be known to conclude that the quadrilateral $\mathrm{PQOR}$ is a square?

i) $\mathrm{OQ}$ and $\mathrm{OR}$ are at an angle of $90°$.

ii) The tangents meet at an angle of $90°$.

In the following figure, $\mathrm{O}$ is the centre of the circle and $\mathrm{PQ}$ and $\mathrm{PS}$ are tangents to the circle at points $\mathrm{Q}$ and $\mathrm{S}$ respectively.

What is the measure of $\angle \mathrm{QRS}$? (Without symbol) Show your work.

Shown below is a circle with centre $\mathrm{O}$ and tangents $\mathrm{PQ}$ and $\mathrm{PR}$.

Using triangles $\mathrm{QOR}$ and $\mathrm{PQR}$, and without doing any extra constructions, prove that the tangents $\mathrm{PQ}$ and $\mathrm{PR}$ are equal in length.

A point is 25 cm from the centre of a circle of radius 15 cm. Find the length of the tangent from the point to 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.

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 $∆\mathrm{PQR}.$