Hello, and welcome to your Class 10 Mathematics lesson on Circles. Today, we will explore one of the most elegant topics in geometry — the circle and its remarkable properties. We will learn about chords, arcs, segments, and the powerful theorems that connect angles at the centre to angles at the circumference. We will also discover what makes a quadrilateral cyclic, and how to apply these results to solve challenging problems. Let us begin.
First, let us establish what a circle actually is. A circle is the locus of a point moving in a plane such that its distance from a fixed point remains constant. This fixed point is called the centre, and the constant distance is called the radius. The perimeter of the circle is called its circumference.
Now, let us understand some special types of circles and their relationships with polygons. Concentric circles are circles that share the same centre but have different radii. Equal circles, or congruent circles, are circles with equal radii.
When a circle passes through all the vertices of a polygon, we call it a circumscribed circle, and the polygon is said to be inscribed in the circle. The centre of this circle is called the circumcentre. Conversely, when a circle touches all the sides of a polygon, it is called an inscribed circle or incircle, and the polygon is said to be circumscribed about the circle.
Let us now turn to chords — the building blocks of circle geometry. A chord is a line segment joining any two points on the circumference of a circle. The chord that passes through the centre is called the diameter, and it is the largest possible chord.
Here is our first important result. A straight line drawn from the centre of a circle to bisect a chord, which is not a diameter, is at right angles to the chord. Conversely, the perpendicular to a chord, from the centre of the circle, bisects the chord. These two results are extremely useful and often used together.
There is a beautiful relationship between the size of a chord and its distance from the centre. The greater the chord, the smaller its distance from the centre, and vice versa. Equal chords are equidistant from the centre, and chords equidistant from the centre are equal.
Another fundamental fact: given three non-collinear points, there exists exactly one circle passing through all three. This is why three points uniquely determine a circle.
Now we move to arcs and segments. An arc is simply a part of the circumference. A chord divides the circumference into two arcs: the minor arc, which is smaller than a semicircle, and the major arc, which is larger.
A segment is the region bounded by an arc and its corresponding chord. The major segment corresponds to the major arc, and the minor segment corresponds to the minor arc.
Here comes one of the most powerful theorems in circle geometry. The angle which, an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circumference.
Let me explain this carefully. Imagine a circle with centre O, and an arc AB. Draw radii OA and OB, forming angle AOB at the centre. Now pick any point C on the remaining circumference, and draw chords AC and BC, forming angle ACB. The theorem states that angle AOB equals twice angle ACB.
The proof relies on isosceles triangles and exterior angle properties. Join CO and extend it to a point D. Since OA equals OC, triangle AOC is isosceles, so angle OAC equals angle OCA. The exterior angle AOD equals angle OAC plus angle OCA, which is twice angle OCA. Similarly, angle BOD equals twice angle OCB. Adding these, angle AOB equals twice angle ACB. This theorem is the gateway to many other results.
From this central theorem, we immediately obtain two corollaries.
First, angles in the same segment of a circle are equal to one another. If two points C and D lie on the same side of chord AB, then angle ACB equals angle ADB. This follows because both angles equal half of angle AOB.
Second, the angle in a semi-circle is a right angle. If AB is a diameter, then angle ACB equals 90 degrees for any point C on the circumference. This is because angle AOB is 180 degrees, so angle ACB is half of that. This result, attributed to Thales, is one of the oldest theorems in mathematics.
Let us work through a quick example to see these theorems in action.
Suppose in a circle, PQ equals PR, and angle PRQ equals 70 degrees. We need to find angle QAR. In triangle PQR, since PQ equals PR, angles PQR and PRQ are both 70 degrees. Therefore, angle QPR equals 180 minus 140, which is 40 degrees. Now, angles QAR and QPR are in the same segment, so they are equal. Thus, angle QAR equals 40 degrees.
Here is another example. Given a cyclic quadrilateral ABCD with angle BAC equal to 67 degrees, find the sum of angle DBC and angle DCB. First, angle BDC equals angle BAC, which is 67 degrees, since they are in the same segment. In triangle DBC, the sum of angles is 180 degrees. So the sum of angle DBC and angle DCB equals 180 minus 67, which equals 113 degrees.
Now we enter the rich territory of cyclic quadrilaterals. A cyclic quadrilateral is one whose vertices all lie on a circle. Points that lie on the same circle are called concyclic points.
The opposite angles of a cyclic quadrilateral are supplementary; that is, they sum to 180 degrees. Angle ABC plus angle ADC equals 180 degrees, and angle BAD plus angle BCD equals 180 degrees.
The proof uses our central angle theorem. Join the centre O to vertices A and C. Arc ABC subtends angle AOC at the centre and angle ADC at the circumference, so angle ADC equals half of angle AOC. Similarly, arc ABC subtends the reflex angle AOC at the centre and angle ABC at the circumference, so angle ABC equals half of the reflex angle AOC. Since angle AOC plus reflex angle AOC equals 360 degrees, the sum of their halves equals 180 degrees.
The converse is equally important: if a quadrilateral has opposite angles supplementary, then it is cyclic.
From this property flows another elegant result. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. If side AB of cyclic quadrilateral ABCD is produced to a point E, then exterior angle CBE equals interior opposite angle ADC. This follows because angle ABC plus angle CBE equals 180 degrees, and angle ABC plus angle ADC equals 180 degrees.
Let us see an application. Suppose angle AOC equals 110 degrees in a cyclic quadrilateral. Then angle ADC equals half of 110, which is 55 degrees. Angle ABC equals 180 minus 55, which is 125 degrees. In triangle AOC, since OA equals OC, angles OAC and OCA are equal, and each equals half of 180 minus 110, which is 35 degrees.
We now turn to some additional important results about chords and angles.
In a circle, equal chords subtend equal angles at the centre of the circle. Conversely, if chords subtend equal angles at the centre, they are equal. This gives us a powerful way to relate chord lengths to central angles.
For a regular polygon inscribed in a circle, the central angle subtended by each side equals 360 degrees divided by the number of sides. So for a square, each central angle is 90 degrees. For a regular pentagon, it is 72 degrees. For a regular hexagon, it is 60 degrees.
The ratio of chord lengths equals the ratio of their corresponding central angles. If chord AB to chord CD is in ratio 7 to 5, then angle AOB to angle COD is also 7 to 5.
Here is an example involving arc lengths. If arcs AB and BC are in ratio 3 to 2, and angle AOB equals 96 degrees, then angle BOC equals two-thirds of 96, which is 64 degrees. Angle CAB equals half of angle BOC, which is 32 degrees. Angle ACB equals half of angle AOB, which is 48 degrees. Since ADBC is cyclic, angle ADB equals 180 minus 48, which is 132 degrees.
Let us prove an interesting property: if two sides of a cyclic quadrilateral are parallel, then the other two sides are equal.
Given cyclic quadrilateral ABCD with AB parallel to DC. Since ABCD is cyclic, angle ABC plus angle ADC equals 180 degrees. Since AB is parallel to DC, angles BAD and ADC are co-interior angles, so they sum to 180 degrees. Therefore, angle ABC equals angle BAD. In triangles BAD and ABC, we have angle BAD equals angle ABC, angle ACB equals angle ADB as angles in the same segment, and AB is common. Thus the triangles are congruent by A.S.A., and AD equals BC by C.P.C.T.C.
An alternative proof uses the centre: join OA, OB, OC, and OD. Since AB is parallel to DC, angle BAC equals angle ACD as alternate interior angles. These angles subtend arcs BC and AD at the circumference. Equal angles at the circumference imply equal arcs, which imply equal chords. Thus AD equals BC.
Let us recap the key takeaways from this lesson.
First, the perpendicular from the centre to a chord bisects the chord, and equal chords are equidistant from the centre.
Second, the angle subtended by an arc at the centre is twice the angle subtended at any point on the remaining part of the circumference.
Third, angles in the same segment are equal, and the angle in a semi-circle is a right angle.
Fourth, opposite angles of a cyclic quadrilateral are supplementary, and the exterior angle equals the interior opposite angle.
Fifth, equal chords subtend equal angles at the centre, and chords are proportional to their central angles.
These five pillars will support your understanding of circle geometry and help you solve even the most challenging problems.
That brings us to the end of our lesson on Circles. I encourage you to practice applying these theorems, drawing diagrams, and explaining your reasoning step by step. Geometry rewards patience and precision. Until next time, keep exploring, keep questioning, and enjoy the beauty of mathematics.