Circles and Geometric Shapes

Draw $\Delta\text{ABC}$ with $\text{AB} = 5\text{ cm}$, $\angle\text{A} = 70^\circ$ and $\angle\text{B} = 60^\circ$. Draw the circumcircle of $\Delta\text{ABC}$. Is the centre inside or outside the triangle?

Draw $\Delta \text{ABC}$ with $\text{AB} = 5 \text{ cm}$, $\angle \text{A} = 100^\circ$, $\text{AC} = 4 \text{ cm}$. Draw the circumcircle of $\Delta \text{ABC}$. Is the centre inside or outside the triangle?

Draw $\Delta \text{ABC}$, with $\text{AB} = 6\text{ cm}$, $\text{BC} = 7\text{ cm}$ and $\text{CA} = 7\text{ cm}$. Draw the circumcircle of $\Delta \text{ABC}$. Let the circumcentre be $O$. Measure $\text{OA}$, $\text{OB}$, $\text{OC}$.

What is the least possible radius of a circle through two points $A$ and $B$?

Show that the triangle formed by a chord and the centre of the circle is isosceles.

Show that if two such isosceles triangles (occurring in the previous question) have equal base length, they are congruent to each other.

Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre of a circle to a chord of the circle bisect the chord?

(Hint: Use Fig. 5.12. You are told that $\angle CMA = \angle CMB = 90^\circ$. You need to show that $AM = BM$.)

An isosceles triangle $ABC$ is inscribed in a circle, with $AB = AC$. Show that the altitude from $A$ to $BC$ passes through the centre of the circle.

Two parallel chords of lengths $6\text{ cm}$ and $8\text{ cm}$ are on opposite sides of the centre of a circle. If the radius of the circle is $5\text{ cm}$, find the distance between the midpoints of the chords.

Use the Baudhāyana–Pythagoras theorem to show why Theorem 6 must be true.

Consider Fig. 5.15. If $CE$ is perpendicular to $AB$, $CH$ is perpendicular to $GH$, and $CE = CH$, show that $AB = GF$.

Solve the previous question using the Baudhāyana–Pythagoras theorem.

Find the length of the chord of a circle where the radius is $7\text{ cm}$ and perpendicular distance is $6\text{ cm}$.

Explain why the following statement is true: If the perpendicular distance of a chord from the centre is $d$ and the radius is $r$, then the chord length is $2\sqrt{r^2 - d^2}$.

In a circle, if the distance of chord $AB$ from the centre is twice the distance of another chord $CD$ from the centre, then can we conclude that $CD = 2AB$? Give reasons for your answer.

In a circle with centre $O$, the central angle $\angle AOB$ is $60^\circ$. If the radius of the circle is $12\text{ cm}$, what is the length of the chord $AB$?

Let $A$ and $B$ be two points on a circle with centre $O$.

(i) Are there points $X, Y$ on the circle, on the same side of $AB$, such that $\angle AXB$ is different from $\angle AYB$?

(ii) Is it true that if $\angle AXB = \angle AYB$, then $X$ and $Y$ lie on the same side of the circle?

(iii) If $\angle AXB = \angle AYB$, and $X$ and $Y$ do not lie on the circle, does the circle through $A$, $B$ and $X$ also pass through $Y$?

Find $x$ in Fig. 5.26.

In a circle, a chord is $5\text{ cm}$ away from the centre. If the radius of the circle is $13\text{ cm}$, what is the length of the chord?

An arc of a circle subtends an angle of $70^\circ$ at the centre. What is the measure of the angle subtended by the arc at a point on the circle?

The diameter of a circle is $26 \text{ cm}$. A chord of length $24 \text{ cm}$ is drawn in the circle. Find the distance from the centre of the circle to the chord.

A circle has a radius of $15\text{ cm}$. A chord is drawn. The distance from the centre of the circle to the chord is $9\text{ cm}$. What is the length of the chord?

Prove that the perpendicular bisector of a chord passes through the centre of the circle.

The diameter of a circle is $AB$. Point $C$ is on the circumference. What is the measure of the $\angle \text{ACB}$? Explain your reasoning.

ABCD is a cyclic quadrilateral inscribed in a circle. If $\angle A$ measures $75^\circ$, what is the measure of $\angle C$? If $\angle B$ measures $110^\circ$, what is the measure of $\angle D$?

Quadrilateral $PQRS$ is inscribed in a circle. If $\angle P = (2x + 10)^\circ$ and $\angle R = (3x - 20)^\circ$, find the value of $x$ and the measures of $\angle P$ and $\angle R$.

The distance of a chord of length $16\text{ cm}$ from the centre of a circle is $6\text{ cm}$. Find the radius of the circle.

A cyclic quadrilateral has sides $5$, $5$, $12$, $12$ units. Find its area.

Consider a cyclic quadrilateral. Without drawing its circumcircle, how can we find out whether the centre of the circumcircle lies inside the quadrilateral or outside? What is the best way of finding out?

When two chords intersect, each of them is divided into two line segments. Show that if the intersecting chords are of equal length, then the line segments of one chord are equal to the corresponding line segments of the other chord.

Draw a circle in which a chord of $6\text{ cm}$ length stands at a distance of $3\text{ cm}$ from the centre.

(Hint: Is it a circumcircle of a suitable triangle?)

Show that rectangle is the only parallelogram that can be inscribed in a circle.

Show that if a rectangle is inscribed in a circle, then the point of intersection of its diagonals must lie at the centre of the circle.

Consider all chords of a circle of a fixed length. What is the shape formed by the midpoints of all these chords?

In a circle with centre $O$, chords $AB$ and $AC$ are congruent. Explain why this statement is true: "The centre of the circle lies on the angle bisector of $\angle BAC$".

Two parallel chords of lengths $10\text{ cm}$ and $24\text{ cm}$ are on the same side of the centre of a circle. The distance between the chords is $7\text{ cm}$. Find the radius of the circle.

A regular hexagon is inscribed in a circle of radius $r$. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.

A quadrilateral $MNOP$ is inscribed in a circle. If $MN$ is a diameter, what can you say about $\angle MOP$ and $\angle MNP$? Explain your reasoning.

Let $ABCD$ be a cyclic quadrilateral. Explain why the exterior angle at any vertex is equal to the interior opposite angle (e.g., $\angle \text{CDE} = \angle \text{ABC}$, where $E$ is a point on the extension of side $CD$).

"There is no chord of a circle that is longer than its diameter." How do you justify this statement?

Let $A$ be any point within a given circle with centre $O$. Show that the shortest chord of the circle that passes through point $A$ is the one that is perpendicular to $OA$.

How would you use the following figure to justify the statement that the angle in a semicircle is $90^\circ$?

In a circle, two chords $CC'$ and $DD'$ are drawn perpendicular to a diameter $AB$. Prove that the segment $MM'$ joining the midpoints of the chords $CD$ and $C'D'$ is perpendicular to $AB$.

How would you use the following figure to justify the statement that the sum of the opposite angles of a cyclic quadrilateral is $180^\circ$?