Circles and Geometric Shapes | EOT

Question 22

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

Solution

Understand the Question

A square of side $s = 2\text{ units}$ contains a 4-petalled flower formed by intersecting circular arcs of radius $r = 2\text{ units}$ .
Each petal is bounded by $2$ quarter-circle arcs, giving a total of $4 \times 2 = 8$ quarter-circle arcs for the flower's perimeter.
The area of the flower is calculated by subtracting the area of the $4$ unshaded corner regions (circular segments) from the total area of the square.

(i) Find the perimeter of the flower.

Step 1 · Calculate the Perimeter of the Flower

The square has side length $s = 2\text{ units}$ . Each arc forming the petals is a quarter circle of radius $r = 2\text{ units}$ . Diagram 1

$\text{Side length of square } (s) = 2 \text{ units}$ $\text{Radius of each arc } (r) = s = 2 \text{ units}$

Length of one quarter-circle arc:

\begin{aligned} \text{Length of one quarter circle arc} &= \dfrac{1}{4} \times 2\pi r \\[0.6em] &= \dfrac{1}{2}\pi r \\[0.6em] &= \dfrac{1}{2}\pi (2) \\[0.6em] &= \pi \text{ units} \end{aligned}

Since the flower consists of $8$ identical arcs: $\text{Perimeter of flower} = 8 \times \pi = 8\pi \text{ units}$

Answer

(i) $8\pi \text{ units}$

(ii) Find the area of the flower.

Step 1 · Calculate Area of Square and Corner Segments

Area of the square:

\begin{aligned} \text{Area of square} &= s^2 \\ &= (2)^2 \\ &= 4 \text{ square units} \end{aligned}

Each unshaded corner region is a segment formed by a sector of angle $90^\circ$ and radius $r = 2\text{ units}$ minus a right isosceles triangle:

\begin{aligned} \text{Area of one sector} &= \dfrac{90^\circ}{360^\circ} \times \pi r^2 \\[0.6em] &= \dfrac{1}{4}\pi (2)^2 \\[0.6em] &= \pi \text{ square units} \end{aligned}

\begin{aligned} \text{Area of this triangle} &= \dfrac{1}{2} \times \text{base} \times \text{height} \\[0.6em] &= \dfrac{1}{2} \times 2 \times 2 \\[0.6em] &= 2 \text{ square units} \end{aligned}

Area of one unshaded corner segment:

\begin{aligned} \text{Area of one segment} &= \text{Area of sector} - \text{Area of triangle} \\ &= \pi - 2 \text{ square units} \end{aligned}

Total area of the $4$ unshaded regions:

\begin{aligned} \text{Total unshaded area} &= 4 \times (\pi - 2) \\ &= 4\pi - 8 \text{ square units} \end{aligned}

Area of the flower:

\begin{aligned} \text{Area of flower} &= \text{Area of square} - \text{Total unshaded area} \\ &= 4 - (4\pi - 8) \\ &= 4 - 4\pi + 8 \\ &= 12 - 4\pi \text{ square units} \end{aligned}

Answer

(ii) $(12 - 4\pi) \text{ square units}$

Common Mistakes

Arc Count Error: The flower has $4$ petals and each petal has $2$ arcs, making a total of $8$ quarter-circle arcs, not $4$ .
Sign Error in Subtraction: When subtracting total unshaded area, forgetting to distribute the negative sign: $4 - (4\pi - 8) = 12 - 4\pi$ , not $-4 - 4\pi$ .
Sector vs. Segment: Using the full sector area instead of the corner segment area (sector minus triangle) when subtracting unshaded regions.

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?

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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 $\angle R$ .

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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 $180^\circ$ ?