When we study Geometry, and look at the Circle, we say that the circle is a round figure, which has its boundary at equal distance from a Point called Centre of the circle. If we look at different parts of the circle we use the terms center, radius, arc, diameter, sector and a Chord. We first talk about the radius of a circle.
If we draw a line from center of the circle to the boundary of the circle,(which we call the circumference of the circle), then the length of this line is called the radius of the circle. The radius remains of fixed length, from where ever we draw it. Sector, arc, radii, diameter are parts of circle.
If we join the two radius, which moves in the opposite direction from the center of the circle is called the Diameter of the circle, which is a part of a circle. We observe that two radius join to form a diameter, so we say that the diameter = radius * 2.
Now we talk about another term related to the circle i.e. chord. We say that the chord is the Line Segment, which starts at one point on the circumference of the circle and ends at another end. There can be infinite number of chords that can be drawn inside the circle. Moreover, we observe that the diameter is also the chord, as it ends at the circumference of the circle.
We also find that the diameter is the longest chord of the circle. If any line is drawn outside the circle, which touches the circle at only one point, then it is called the Tangent drawn on the circle. Any line drawn from the centre of the circle to the Tangent is always perpendicular. So, we say that the radius is perpendicular to the circle.
Every geometric shape like Circle, triangle, rectangle, square etc has one center and when we rotate any geometric shape around its center, its orientation remains same means length of any straight line
from center to its edge remains same, like we have a circle, whose radius is equal to 7 inch, then from center Point to each point of circle, the length of Straight Line is equals to 7 inch because it behave like a Radius of Circle and this center is called as a centre of a circle
. For finding the centre of circle, we use following methods.
Method 1: If two edge points of Diameter is given like we have a circle, whose two diameter edge points are (p, q) and (h, k), then Center of Circle C (x, y) is equals to,
x = (p + h) / 2 and y = (q + k) / 2,
These two ‘x’ and ‘y’ points are called as a center of circle in this situation.
Suppose we have a diameter, whose edge points are A (4, 7) and B (6, 9), then center of circle–
x = (4 + 6) / 2,
= 10 / 2,
y = (7 + 9) / 2,
= 16 / 2,
So, center point of circle x = 5 and y = 8 means C (5, 8) is a center of circle.
Method 2: If a circle equation is given like–
(x – a)2
+ (y – b)2
Then here value of ‘a’ and ‘b’ is behaving like a center of circle and value of center of circle is equals to C (a, b).
Suppose, we have a circle equation–
(x – 5)2
+ (y – 7)2
Then center point of circle in this situation are x = 5 and y = 7 or C (5, 7) is the center of circle.
In Geometry, as we know that we study about different shapes. Starting with the basic elements, viz., point, line & plane, some shapes might be rectilinear while some others curved; some might be plane figures & some others are solids. In plane figures some might be 2 dimensional while some others might be 3 dimensional. In spite of all such classifications, we start geometry with a Point which has neither length nor breadth, nor height whereas the next proceeding element, i.e., line is a 1 dimensional concept which has only length.
As we just mentioned that some of the shapes might be rectilinear while others curved, in curved figures we will learn about Circle & its chord. A circle is a closed curved figure which is drawn by joining a number of points which are equidistant from a fixed point. This fixed point is called the Centre of the circle & is denoted by ‘O’. The distance from the centre to any point on the circle is called the radius, ‘r’ of the circle.
Now let us know about the chord of a circle. As stated just above that a circle is formed by joining a number of points, these points when joined form a closed curve which is the boundary of the circle. Now if we join any one point on this curve to the centre of the circle, it is termed as the radius of the circle. But if we join any two points on the curve or the boundary of the circle, it is called the chord of the circle. In simpler words, a chord is a line segment joining any two points on a circle, example: if we take two points A & B on the circle & join them, we say that AB is the chord of the circle. This is all about chords of a circle.
We are already familiar with Circles & some terms associated with them. Let us recall that a Circle is a closed curve formed by joining all the points which are equidistant from a fixed Point, ‘O’ in its Centre. This fixed point is the centre of the circle.
In this session, we will learn about a new term associated with circles, i.e., sector of a circle. To understand this concept, let us recall here what the terms radius & arc of a circle Mean. By radius of a circle, we mean the line segment joining the centre of the circle to any point on its boundary. As all the points on the circle are equidistant from its centre, thus radius of a circle is same irrespective of the point taken on a particular circle. Also, the arc of a circle is a part of its curve or the boundary.
Sector of circle
is the area enclosed by an arc & the two radii joining the end points of the arc to the centre of the circle.
With the help of any given arc & the radii we can identify two sectors of a circle. One is the sector covered by the smaller part of the circle, i.e., the Minor Arc
& it gives us the minor sector. Whereas, if we see the other side of the same arc, i.e., the bigger part or the Major Arc
, in other words, we get the major sector. Thus, the two radii & the same two points on the circle defining the arc, give us two different sectors of the circle, minor sector & the major sector .
Circle can be defined as closed round shaped figure. Diameter
is one of the terminologies that we use for circle.
Diameter of circle is defined as a line that passes from center and touches two points on boundary of circle. Often the word diameter is used to refer line itself. Diameter
is twice of radius and is also called as Chord. Chord is defined as any line that joins two points on a circle. So, we can say that diameter is a chord which passes through center of circle and it is also the longest possible chord of circle.
If we have to calculate area or circumference of circle then we need to diameter or radius. As area of circle
is given as:
Area = πr2
Here, 'r' is the radius or we can say half of diameter and 'π' is constant and its value is 3.14 or 22 / 7. If diameter or radius is known then we can easily determine Area of a Circle. Similarly, circumference of a circle is given as:
Circumference = 2 π r,
Here again 'π' is constant and 'r' is radius. So if radius or diameter is given then one can easily calculate area and circumference of circle by putting values in their respective formulas.
As we know that in mathematics Geometry is based on rules, facts, givens, and proofs to find various conclusions.
Now before knowing the meaning of segment of chords we need to learn few terms which are related to Circle like Tangent, chord, radius, circumference, arc etc. So the related terms are
1. Tangent: - any line which is perpendicular to radius and lies outside the circle and also touches the circle at one Point is called tangent and tangent never intersect circle.
2. Chord: - the Diameter of circle is the longest Chord in any circle and the distance from one point to other inside the circle are called chord.
3. Radius: - the distance from central point of circle that is Centre point to any point on circumference of circle are called radius.
4. Circumference: - if we measure boundary of any circle is called circumference of circle.
5. Arc: -if we cut a portion from circumference of circle weather it is large or small it is called arc.
6. Origin: - it means the centre of circle or central point is called origin.
These are the Basic Terms which we used in circle so that while proving any theorem we used these terms to prove the theorems.
Now we are going to explain the meaning of segment of chords
If we have a circle we draw two chords in the same circle in such a manner that both intersect each other so both the chords are divided into two parts. It means we have total four parts so these parts are called segment of chords.
Now we have a chord rule and this rule says that if two chords intersect then the product of one chord segment is equal to the product of second chord segment.
Now suppose that we have a circle whose origin is O and we draw two chords AB and CD and both the chords intersect each other at point H and also the length of AB is 8 cm and CD is 7 cm now when they intersect each other AB chord divided into AH and HB which is 6cm and 2cm respectively and same for second chord which is divided into CH and HD which is 3cm and 4cm respectively now if we multiply the segments of both the chords then they are equal so
AB = CD
AH × HB = CH × HD
6 × 2 = 3 × 4
12 = 12
So the segment of chords is the parts of chords which we get after intersecting of two chords.