Circles and Geometric Shapes | EOT

Question 19

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.

Solution

Understand the Question

A large rectangle with an area of $72 \text{ cm}^2$ is formed by fitting $9$ identical small rectangles together.
Let the length of each small rectangle be $l$ and the width be $w$ .
The top row consists of $4$ rectangles arranged horizontally (length along the width of the large rectangle), giving a width of $4l$ and height $w$ .
The bottom row consists of $5$ rectangles arranged vertically (width along the width of the large rectangle), giving a width of $5w$ and height $l$ .
Since both rows form the same large rectangle, their total widths are equal: $4l = 5w$ .
We use the given area to find $l$ and $w$ , and then calculate the perimeter of one small rectangle using $P = 2(l + w)$ .

Step 1 · Define dimensions and relate length and width

Let the length of each small rectangle be $l$ and the width be $w$ . Diagram 1

Equating the total width of the top row and bottom row $4l = 5w$

$w = \dfrac{4}{5}l$

Step 2 · Set up area equation and solve for dimensions

The dimensions of the large rectangle are:

$\text{Total width} = 4l$
$\text{Total height} = w + l$

Given that the area of the large rectangle is $72 \text{ cm}^2$

\begin{aligned} \text{Area} &= \text{Total width} \times \text{Total height} \\ 72 &= 4l(w + l) \end{aligned}

Substitute $w = \dfrac{4}{5}l$

\begin{aligned} 72 &= 4l\left(\dfrac{4}{5}l + l\right) \\[0.6em] &= 4l\left(\dfrac{4l + 5l}{5}\right) \\[0.6em] &= 4l\left(\dfrac{9l}{5}\right) \\[0.6em] 72 &= \dfrac{36l^2}{5} \end{aligned}

Solve for $l$

\begin{aligned} 36l^2 &= 72 \times 5 \\[0.6em] 36l^2 &= 360 \\[0.6em] l^2 &= \dfrac{360}{36} = 10 \\[0.6em] l &= \sqrt{10} \text{ cm} \end{aligned}

Now find width $w$ $w = \dfrac{4}{5}l = \dfrac{4}{5}\sqrt{10} \text{ cm}$

Step 3 · Calculate perimeter of each small rectangle

Perimeter of one small rectangle is $P = 2(l + w)$

\begin{aligned} P &= 2\left(\sqrt{10} + \dfrac{4}{5}\sqrt{10}\right) \\[0.6em] &= 2\left(\dfrac{5\sqrt{10} + 4\sqrt{10}}{5}\right) \\[0.6em] &= 2\left(\dfrac{9\sqrt{10}}{5}\right) \\[0.6em] &= \dfrac{18\sqrt{10}}{5} \text{ cm} \end{aligned}

Answer

$\dfrac{18\sqrt{10}}{5} \text{ cm}$

Common Mistakes

Equating the Wrong Sides: Incorrectly setting $5l = 4w$ instead of $4l = 5w$ by mixing up the orientations of the top and bottom rows.
Individual Area Shortcut: You can also find the area of one small rectangle directly as $\dfrac{72}{9} = 8 \text{ cm}^2$ , so $l \times w = 8$ . Substituting $w = \dfrac{4}{5}l$ gives $l \times \dfrac{4}{5}l = 8 \implies l^2 = 10$ , which is a faster alternative method.
Perimeter vs. Semi-perimeter: Forgetting to multiply by $2$ in the perimeter formula $P = 2(l + w)$ .

More questions in EOT

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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 $180^\circ$ ?