Circles and Geometric Shapes | EOT

Question 11

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?

Solution

Understand the Question

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on a single circle (the circumcircle).
The centre of this circumcircle (the circumcentre) is equidistant from all four vertices.
Geometrically, the circumcentre is the common point of intersection of the perpendicular bisectors of the sides of the quadrilateral.
Therefore, we can determine the exact position of the circumcentre (inside, on, or outside the quadrilateral) without drawing the full circle by constructing the perpendicular bisectors of its sides.

Step 1 · Use Perpendicular Bisectors of Sides

The circumcentre $O$ is equidistant from all vertices ( $OA = OB = OC = OD = R$ ).

By geometric definition, the locus of points equidistant from two points is the perpendicular bisector of the segment joining them. Thus, the circumcentre must lie on the perpendicular bisector of every side of the quadrilateral.

Step 2 · Step-by-Step Construction Method

The best and most reliable method to locate the circumcentre without drawing the circle is:

Draw the perpendicular bisectors of any two adjacent sides (e.g., sides $AB$ and $BC$ ).
Mark their point of intersection as $O$ .
Check the position of point $O$ $O$ relative to the quadrilateral:
- Inside: If $O$ lies entirely within the interior region of quadrilateral $ABCD$ , the centre lies inside.
- On the boundary: If one of the opposite angle pairs is $90^\circ$ , the diagonal opposite to the right angle is a diameter, and $O$ is the midpoint of that diagonal (lies on the boundary/diagonal).
- Outside: If $O$ falls outside the interior region of $ABCD$ , the centre lies outside.

Step 3 · Angle-Based Check

Alternatively, consider the diagonals $AC$ and $BD$ :

If the opposite angles of the cyclic quadrilateral are right angles ( $\angle B = \angle D = 90^\circ$ or $\angle A = \angle C = 90^\circ$ ), the diagonal acts as the diameter of the circumcircle, and the circumcentre is the midpoint of that diagonal.
If any chord/side subtends an angle greater than $90^\circ$ in the major arc containing the centre, the centre falls outside the quadrilateral across that particular side.

Answer

The best way is to find the point of intersection of the perpendicular bisectors of any two adjacent sides. If this intersection point lies within the interior of the quadrilateral, the centre is inside; if it lies outside the boundary, the centre is outside.

Common Mistakes

Confusing Angle Bisectors with Perpendicular Bisectors: The incenter is found using angle bisectors, whereas the circumcentre is found using perpendicular bisectors of the sides.
Assuming the Centre is Always Inside: Just like with obtuse triangles, if the cyclic quadrilateral is stretched such that one side lies beyond the diameter, the circumcentre can lie outside the quadrilateral.
Intersection of Diagonals: The point where the diagonals intersect is generally not the circumcentre (it only coincides with the circumcentre if the quadrilateral is a rectangle or square).

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

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 $180^\circ$ ?