Circles and Geometric Shapes | EOT

Question 17

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 $\angle BAC$ ".

Solution

Understand the Question

Let each circle have a radius $r$ , which means the diameter of each circle is $2r$ .
The height of the rectangle is equal to the diameter of one circle, and the length is equal to the sum of the diameters of all circles placed side by side.
To find the fraction of the rectangle covered by the circles, calculate the ratio: $\text{Fraction} = \dfrac{\text{Total Area of Circles}}{\text{Area of Rectangle}}$

(i) Fig. 6.45: What fraction of the rectangle is covered by the circles?

Step 1 · Find Dimensions and Areas for Fig. 6.45

Let the radius of each circle be $r$ . Then the diameter of each circle is $2r$ . Diagram 1

Dimensions of the rectangle: $\text{Height} = 2r$ $\text{Length} = 3 \times (2r) = 6r$

Calculate the areas: $\text{Area of 3 circles} = 3 \times \pi r^2 = 3\pi r^2$ $\text{Area of rectangle} = (6r) \times (2r) = 12r^2$

Calculate the fraction covered:

\begin{aligned} \text{Fraction} &= \dfrac{\text{Area of circles}}{\text{Area of rectangle}} \\[0.6em] &= \dfrac{3\pi r^2}{12r^2} \\[0.6em] &= \dfrac{3\pi}{12} \\[0.6em] &= \dfrac{\pi}{4} \end{aligned}

Answer

(i) $\dfrac{\pi}{4}$

(ii) Fig. 6.46: What fraction of the rectangle is covered by the circles?

Step 1 · Find Dimensions and Areas for Fig. 6.46

Let the radius of each circle be $r$ . Then the diameter of each circle is $2r$ . Diagram 2

Dimensions of the rectangle: $\text{Height} = 2r$ $\text{Length} = 4 \times (2r) = 8r$

Calculate the areas: $\text{Area of 4 circles} = 4 \times \pi r^2 = 4\pi r^2$ $\text{Area of rectangle} = (8r) \times (2r) = 16r^2$

Calculate the fraction covered:

\begin{aligned} \text{Fraction} &= \dfrac{\text{Area of circles}}{\text{Area of rectangle}} \\[0.6em] &= \dfrac{4\pi r^2}{16r^2} \\[0.6em] &= \dfrac{4\pi}{16} \\[0.6em] &= \dfrac{\pi}{4} \end{aligned}

Answer

(ii) $\dfrac{\pi}{4}$

Common Mistakes

Using Radius Instead of Diameter: Taking the height as $r$ instead of the diameter $2r$ , leading to an incorrect rectangle area.
Omitting the Number of Circles: Forgetting to multiply the single circle area $\pi r^2$ by the total number of circles ( $3$ or $4$ ).
General Invariance: Notice that for any number $n$ of identical circles packed in a single row inside a fitting rectangle, the fraction covered is always $\dfrac{n \pi r^2}{(n \cdot 2r)(2r)} = \dfrac{\pi}{4}$ .

More questions in EOT

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.

Q10

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

Q11

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?

Q12

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.

Q13

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?)

Q14

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

Q15

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.

Q16

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

Q17

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 $\angle BAC$ ".

Q18

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.

Q19

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.

Q20

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.

Q21

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 $CD$ ).

Q22

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

Q23

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 $OA$ .

Q24

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

Q25

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 $AB$ .

Q26

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