Circles : Construction and Properties

Like the geometric figure of a triangle, study of a circle also forms one of the most basic aspects in geometry.  In the present chapter we will see how to construct a circle and get to know some of the properties of a circle.    

What we will study in this chapter : 

1. Construction of a circle 
2. Some properties a circle  

1. Construction of a circle
Place a point O on a piece of paper. Measure a length of say 5 cm on the compass. Place the steel needle of the compass on point O and rotate the compass all around. You will obtain a circle centered on point O. Point O is called the center of the circle. The circular line is called the circumference of the circle.  All points on the circumference are equidistant from the center O. This distance is called the radius of the circle. In the present example, the radius of the circle is 5 cm.

The diameter of the circle is the length of the segment that joins two points on the circumference, but passes through O.  

Segment AB is the diameter of the circle, Segment  AO = Segment BO = radius of the circle = r Diameter = 2 * radius = 2r

One very fundamental quantity called pi () is associated with a circle. Pi is defined as the ratio of the circumference to the diameter of a circle.
The circumference of a circle = 2 r
Pi is a constant quantity and is given by

   =   22/7   = 3.142

Another important point regarding a circle is that its angular measure is 360ƒ. This means that  if you put the protractor on the center O, you will notice that the complete circumference subtends an angle of 360ƒ at the center.

2. Some properties of a circle  
We have already seen what are radius, diameter, circumference and center of a circle. Now we will study what are the area, arc, sector, chord, tangent and secant of a circle.

Area of a circle =  r²  =     d²/4     (because  d = 2r )        

Arc of a circle : Any  portion of the circumference is called the arc of the circle. In the adjoining figure, AXB is an arc of a circle, Similarly AYB is also an arc of a circle.  Join AO and BO. AOB is the angular measure of arc AXB. Since  AOB < 180ƒ, arc AXB is known as the minor arc. Measure of an arc is also written as  m (arc AXB). Arc AYB is known as the major arc and m (arcAYB) = 360ƒ - m(arc AXB). The angle subtended by a major arc at the center O is always > 180ƒ.

If A and B are two end points of a diameter, then the m (arc AXB) = m (arc AYB) = 180ƒ

If  AOB = ƒ, then the length of an arc is /360 of the circumference. That is the entire circumference subtends an angle of 360ƒ at the center O, and the arc AXB subtends and angle at O, then the

length of arc AXB =  ƒ / 360^o   *   2 r

Sector of a circle : The area bounded by an arc and the two radii joining the arc, is known as the sector of a circle. 

Arc AXB along with radii AO and BO form a sector of a circle.  The sector is denoted as O-AXB.

If  AOB = ƒ, then the

area of the  sector  =  ƒ / 360^o   *   r²

Chord of a circle : A segment whose both the end points coincide or lie on the circumference of the circle, is known as the chord of a circle.

In the circle shown above, only segment AB have both the end points lying on the circumference. Hence seg AB is a chord of a circle.  For seg CD only one point C lies on the circumference, hence seg CD is not a chord of the circle. Same is true with seg PQ and seg EF.

Diameter is the largest chord of the circle.

Secant of a circle : A line, which intersects the circumference of the circle in two distinct points, is known as the secant of a circle.

Line l does not intersect the circle hence it is not a secant. Lines m and n satisfy the condition of a secant as they are intersecting with the circle at two distinct points. Hence lines m and n are secants of the circle. Line k is intersecting at only point; this is a special case of a secant. It is worthwhile to note that the secant and the circle lie in the same plane.

Tangent of a circle : A line, which lies in the plane of the circle and intersects with it in just one point only, is known as the tangent of a circle. In the above circle, line k is a tangent of the circle. The intersection point of line k with the circle is A. Radius OA will subtend a right angle with respect to line k.  

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