Circles and Geometric Shapes | EOT

Question 12

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.

Solution

Understand the Question

We are given a circle where two equal chords, $AB$ and $CD$ , intersect at a point $E$ inside the circle.
We need to show that the line segments of one chord are equal to the corresponding line segments of the other chord: $AE = CE$ and $EB = ED$ .
Key Geometric Properties Used:
- Equal chords are equidistant from the centre of the circle.
- The perpendicular from the centre to a chord bisects the chord.
- Right-hypotenuse-side (RHS) congruency criterion.

Step 1 · Set Up Given, To Prove, and Construction

Let a circle with centre $O$ have two equal chords $AB$ and $CD$ intersecting at point $E$ inside the circle.Given: $AB = CD$

To Prove:

$AE = CE$
$EB = ED$

Construction: Draw $OM \perp AB$ and $ON \perp CD$ . Join $OE$ .

Step 2 · Prove Congruence of Triangles OME and ONE

In right-angled triangles $\Delta OME$ and $\Delta ONE$ :

\begin{aligned} \angle OME &= \angle ONE = 90^\circ && \text{(By construction)} \\ OE &= OE && \text{(Common hypotenuse)} \\ OM &= ON && \text{(Equal chords are equidistant from the centre)} \end{aligned}

By RHS congruence criterion: $\Delta OME \cong \Delta ONE$

By CPCT (Corresponding Parts of Congruent Triangles): $ME = NE \quad \dots (1)$

Step 3 · Prove the Segments Are Equal

Since the perpendicular from the centre to a chord bisects the chord:

AM = MB = \dfrac{1}{2} AB

CN = ND = \dfrac{1}{2} CD

Since $AB = CD$ , their halves are also equal: $AM = CN \quad \dots (2)$ $MB = ND \quad \dots (3)$

Adding equation $(1)$ and equation $(2)$ :

\begin{aligned} AM + ME &= CN + NE \\ AE &= CE \quad \dots (4) \end{aligned}

Subtracting equation $(4)$ from $AB = CD$ :

\begin{aligned} AB - AE &= CD - CE \\ EB &= ED \end{aligned}

Answer

Hence proved, $AE = CE$ and $EB = ED$ .

Common Mistakes

Confusing Bisected Halves with Chord Segments: Assuming that $E$ is the midpoint of the chords. The chords intersect at an arbitrary point $E$ , so $AE \ne EB$ in general; rather, the perpendiculars $OM$ and $ON$ bisect the chords at $M$ and $N$ .
Missing the Equidistant Theorem: Forgetting the justification that $OM = ON$ holds because chords of equal length are equidistant from the centre.

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

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 $180^\circ$ ?