Circles

Circle:

A circle is the collection of all those points in a plane whose distance from a fixed point remains constant.
The fixed point is called the centre of the circle, and the constant distance is known as the radius of the circle.

Let O be the centre of a circle of radius r, and let P be a point on the circle. Then, the line segment OP is the radius of the circle, if Q is another point on the circle then the line segment OQ is also a radius of the circle. Clearly. OP = OQ = r.

In general, all the radii of a circle are equal.

Interior and Exterior of a circle:

Let us consider a circle with centre O and radius r. The circle divides the plane containing it into three parts:
1) The part of the plane, consisting of those points P for which OP < r is called the interior of the circle.
(2) The part of the plane, consisting of those points P for which OP = r, is the circle itself.
3) The part of the plane, consisting of those points P for which OP > r is called the exterior of the circle.
Clearly, the circle is the boundary of its interior.

Circular Region:

The part of the plane consisting of the circle and its interior is called the circular region.

Diameter:

A line segment passing through the centre of a circle and having its end points on the circle is called a diameter of the circle.
Clearly, diameter = 2 x (radius).

An infinite number of diameters of a circle can be drawn, Clearly, all the diameters of a circle are concurrent. The centre is their point of concurrence.

Chord:

A chord of a circle is a geometric line segment whose endpoints both lie on the circle.

AB is a chord of the circle with centre O. CD is also a chord but, it is a special chord that passes through the centre of the circle.
A chord that passes through the circle’s centre point is the circle’s diameter.

Diameter of a circle is its largest chord.

Among properties of chords of a circle are the following:
1) Chords are equidistant from the centre only if their lengths are equal.
2) A chord that passes through the centre of a circle is called a diameter, and is the longest chord.
3) If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP x PB = CP x PD (power of a point theorem).
The area that a circular chord “cuts off” is called a circular segment.

Secant of a Circle:

A secant line of a circle is a line that (locally) intersects two points on the circle.

AB is a secant of the circle with centre O.

Circumference of a Circle:

Circumference is the linear distance around the edge of a closed curve or circular object. The circumference of a circle is of special importance to geometric and trigonometric concepts. However circumference may also describe the edge of elliptical closed curve. Circumference is a special case of perimeter in that the perimeter is typically around a polygon while circumference is around a closed curve.

Draw a circle on a thermocol sheet. Fix pins along the circle at close distances.
Wind a thread around the pins and find the circumference of the circle by measuring the length of the thread.
Now, measure the diameter of the circle.

Calculate \(\frac{Circumference}{Diameter}\)

Repeat the procedure with circles of different radii.
You will find that \(\frac{Circumference}{Diameter}\) = 3(approximately) for all the circles.

Segments of a Circle:

A chord AB of a circle divides the circular region into two parts. Each part is called a segment of the circle.

The segment containing the centre of the circle is called the major segment,while the segment not containing the centre is called the minor segment of the circle.

Semicircle:

The end points of a diameter of a circle divide the circle into two equal parts: each part is called a semicircle.

A diameter of a circle divides the circular region into two equal parts; each part is called a semi circular region.

Arc of a Circle:

A continuous piece of a circle is called an arc of a circle.

Consider a circle with centre O and radius r. Let A and B be two points of the circle such that the line segment AB is not a diameter of the circle. Then points A and B divide the circle into two parts one smaller than the other. Each part is an arc of the circle. Points A and B are common to both the arcs. Also, the arcs lie on opposite sides of the chord AB. The smaller arc lies on the side of AB opposite to that of the centre O.

Major and Minor Arcs:

Of the two parts of a circle determined by a chord AB, the smaller part is called the minor arc and the greater part is called the major arc of the circle.

Sector of a Circle:

The area bounded by an arc and the two radii joining the end points of the arc with the centre is called a sector.
If the sector is formed by a major arc, it is called a major sector. If the sector is formed by a minor arc, it is called a minor sector.

OACB is a minor sector, while OADB is the major sector.

Concentric Circles:

Two or more circles with the same centre are called concentric circles.

Let OP  and OQ be radii of the two circles as shown in fig. We can see that their centre’s are same while the radii are different. Such type of circles are called as concentric circles.