Hello, and welcome to today's mathematics lesson. In this session, we will explore one of the most fundamental and beautiful shapes in geometry — the circle. We will begin with its definition, then examine its parts and properties, and finally discover some remarkable theorems about chords and arcs. Let us begin our journey into the world of circles.
A circle is defined as the locus of a point which moves in a plane in such a way that its distance from a fixed point, in the same plane, always remains constant. This constant distance is called the radius. Equivalently, it is the closed curve obtained by joining all points in a plane at the same fixed distance from a fixed point.
The perimeter of a circle is called its circumference. A chord is any straight line segment joining two points on the circumference. The diameter is a special chord that passes through the centre — it is the longest possible chord in any circle. Since the diameter stretches from one side of the circle to the other through the centre, it equals twice the radius. So if r represents the radius, then the diameter equals 2r.
Now let us understand how points relate to a circle. Consider a circle with centre O and radius r. A point lying in the same plane is called an exterior point if its distance from the centre is greater than r. It is called an interior point if its distance from the centre is less than r. And if a point's distance from the centre exactly equals r, then that point lies on the circumference of the circle.
Two or more circles sharing the same centre but having different radii are called concentric circles. Circles with equal radii are called equal circles or congruent circles.
When a circle passes through all the vertices of a polygon, we call it a circumscribed circle, and its centre is the circumcentre. The polygon is then said to be inscribed in the circle. Conversely, when a circle touches all the sides of a polygon, it is called an inscribed circle or in-circle, with its centre being the incentre.
Let us now examine three important regions related to circles: arcs, segments, and sectors.
An arc is simply a part of the circumference. When a chord divides the circumference, it creates two arcs. The smaller one is called the minor arc, and the larger one is called the major arc. When both arcs are equal, each is called a semicircle.
A segment is the region bounded by an arc and its chord. The smaller region is the minor segment, and the larger is the major segment. Importantly, the centre of the circle always lies in the major segment.
A sector is the region bounded by an arc and the two radii joining the centre to the endpoints of that arc. Just like with arcs, we have minor and major sectors. For a semicircle, both segments are equal and both sectors are equal.
Now we come to some beautiful theorems about chords.
Theorem 22 states: 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.
Imagine a circle with centre O and chord AB. If line OC bisects this chord at point C, then OC is perpendicular to AB. The proof uses triangle congruence: we join OA and OB, which are both radii and therefore equal. Since OC is common to both triangles OAC and OBC, and AC equals BC by the bisection, the triangles are congruent by SSS. This makes angles OCA and OCB equal, and since they form a straight line summing to 180 degrees, each must be 90 degrees.
Theorem 23 is the converse: The perpendicular from the centre of a circle to a chord bisects the chord.
If OP is perpendicular to chord AB at point P, then AP equals BP. Here we use RHS congruence: right angles at P, hypotenuse OA equals hypotenuse OB as radii, and OP is common. Therefore, the triangles are congruent, making AP equal to BP.
From these theorems, an important relationship emerges: the greater the chord, the smaller its distance from the centre, and vice-versa.
Theorem 24 states: Equal chords of a circle are equidistant from the centre.
Given chords AB and CD of equal length, with perpendiculars OP and OQ from centre O, we prove OP equals OQ. Since perpendiculars from the centre bisect chords, BP equals half of AB and DQ equals half of CD. With AB equal to CD, we get BP equal to DQ. Triangles OPB and OQD are congruent by RHS, so OP equals OQ.
Theorem 25 reverses this: Chords equidistant from the centre are equal.
Theorem 26 is remarkable: There is one and only one circle, which passes through three given points not in a straight line.
Given three non-collinear points A, B, and C, join AB and BC. Draw their perpendicular bisectors, which intersect at point O. Since every point on a perpendicular bisector is equidistant from the segment's endpoints, O is equidistant from A, B, and C. Thus a circle with centre O and radius OA passes through all three points. Since two lines intersect at only one point, this circle is unique.
This theorem also tells us that the perpendicular bisector of any chord passes through the centre, and the perpendicular bisectors of any two chords intersect at the centre.
Let us work through a practical example.
Suppose diameter CD meets chord AB at point E, with AE and BE each equal to 4 centimetres, and CE equal to 3 centimetres. We need to find the radius.
Let the radius be r centimetres. Then OB and OC both equal r, so OE equals r − 3. Since E is the midpoint of AB, angle OEB is 90 degrees. Applying Pythagoras in triangle OEB: r² = (r − 3)² + 4², which gives r² = r² − 6r + 25. Simplifying, 6r = 25, so r = 25/6 or 4 1/6 centimetres.
Here is another example. A chord of 48 centimetres is 10 centimetres from the centre. We first find the radius using Pythagoras: r² = 10² + 24², which is 676, so r = 26 centimetres. Now for a 20 centimetre chord, half of it is 10 centimetres. Using Pythagoras again: OQ² = 26² − 10², which is 576, giving OQ = 24 centimetres. Notice how the shorter chord is farther from the centre, confirming our theorem.
Now we turn to properties of arcs and their relationships with chords.
Equal arcs of a circle subtend equal angles at the centre, and conversely, if two arcs subtend equal angles at the centre, they are equal. Furthermore, equal arcs cut equal chords, and equal chords cut equal arcs.
Theorem 27: If two arcs of the same circle subtend equal angles at the centre, they are equal.
Given arcs APB and CQD with ∠AOB equal to ∠COD, we draw chords AB and CD. Triangles AOB and COD are congruent by SAS, since two radii and the included angle are equal. Therefore chords AB and CD are equal, which means arcs APB and CQD are equal.
Theorem 28 is the converse: Equal arcs subtend equal angles at the centre. Equal arcs give equal chords by definition, and SSS congruence of the triangles gives equal angles at the centre.
These relationships extend by proportion: if one arc is twice another, its central angle is twice as large, and the ratio of arcs equals the ratio of their central angles.
For example, a regular octagon has central angle 360°/8 = 45°, and a regular hexagon has 360°/6 = 60°.
Let us recap the key takeaways from this lesson.
First, a circle is the locus of points at fixed distance from a centre, with radius r and diameter 2r. Second, the perpendicular from the centre to a chord bisects the chord, and conversely, the line from the centre to the midpoint of a chord is perpendicular to it. Third, equal chords are equidistant from the centre, and chords equidistant from the centre are equal. Fourth, exactly one circle passes through any three non-collinear points. Fifth, equal arcs subtend equal angles at the centre and cut equal chords. Sixth, for a regular n-sided polygon inscribed in a circle, each side subtends angle 360°/n at the centre.
That concludes our exploration of circles. You have learned fundamental definitions, discovered elegant theorems about chords and their distances from the centre, and explored the beautiful relationships between arcs, chords, and central angles. These properties form the foundation for much of advanced geometry. Keep practicing, stay curious, and I look forward to our next mathematical journey together.