Sunday, March 16, 2014

Geometry(Part-3) : Circles and their properties

Posted by Jasleen Behl
I have already discussed some part of geometry in my previous sessions. Both are related to triangles and their properties. Also, I have discussed some Important Questions of geometry. Today I will discuss circles and its properties in exam point of view. Firstly I will tell you some terms related to figure of circle.

Basic Terminologies

  • Centre: 'O' is a centre of the circle i.e. fixed point in the circle.
  • Radius: OM is a radius of circle i.e.from centre to the boundary of circle.
  • Diameter: Double of radius. In figure, XY is a diameter.
  • Circumference: 2`pi`r i.e. distance around the circle.
  • Secant: A line which intersect the circle at two points. In figure, line PQ intersect circle at AB, So PQ is a secant.
  • Tangent: Any line which touches the circle at only one point is tangent of circle. TU is a tangent of circle.
  • Chord: Line whose end points touches the circle . RS is a chord of circle.
  • Segment: Chord divides the circle into two parts. one major and other minor. Both parts are known as Segments. In figure, yellow region is a minor segment. 

Important Results of Circles

Area of Circle

If r is radius of circle, then area of circle = `pi r^2`

Example1: If area of circle is 314, then what will be the radius of circle?
Solution: Area = `pi r^2`
⇒ 200 = `pi r^2`
⇒ r^2 = `314/3.14` (`pi` = 3.14) 
⇒ r = 10

Secants

In a figure given below, PA is a tangent and PRS is a secant, then
PR `times` PS = PA^2 (Result for one tangent and one secant)
If there are two secants, for example in figure, PRS and PTU are secants, then result followed will be:
PT `times` PU = PR `times` PS

Angles of same segment

Yellow part is a segment of circle.
Always remember that:
  • Angles subtended by same segments are always equal. In figure, angle P and Q are equal, as they are subtended by same segment.
  • Angle subtended by segment at the centre is twice the angle subtended at the boundary Angle O = 2 angle P = 2 angle Q.
 Study the following figure:




Above figure shows that, If OC is a perpendicular to AB, C will be the mid-point of AB
i.e. AC = CB and vice-versa.

Alternate segment theorem


If AB is tangent, then alternate angles will be equal in given figure.

I hope you understand the above important results. Will soon update more articles on Geometry. Thanks for visiting.
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