NCERT Solutions for Chapter: Circles, Exercise 1: Exercise

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 $\mathrm{O}$ 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.

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.

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 $90°$.

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?

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 $∆\mathrm{PXY}$.

Here is a circle with centre $O$.

Manu wants to draw a tangent $RS$  to the circle.
What is the number of points at which the line $RS$ will meet the circle?

Anu says, 'A circle of radius $10 \mathrm{cm}$ can have $100$ tangents.' Is Anu correct? Give a reason to justify your answer.