Circle

CIRCLE

Definition:-  Two – dimension shape made by drawing a curve that is always the same distance from a center

OR

A round plane figure whose circumference (boundary) consists of point equidistance from the fixed point (center)

OR

The set of points in a plane which are equidistant from a fixed point in the plane is called a circle.

Some terms related with a circle.

     1)     The fixed point is called the centre of the circle.

     2)     The segment joining the centre of the circle and a point on the circle is called a radius of the circle.

     3)     The distance of a point on the circle from the centre of the circle is also called the radius of the circle.

     4)     The segment joining any two points of the circle is called a chord of the circle.

    5)    A chord passing through the centre of a circle is called a diameter of the circle. A diameter is a largest chord of the circle.

                             

Theorem 1:- A perpendicular drawn from the centre of a circle on its chord bisects the chord.

Chord bisector

Given:-  seg AB is a chord of a circle with centre O. seg OP ⊥ chord AB

To prove:-  seg AP ≅ seg BP

Proof:- :  Draw seg OA and seg OB

In Δ OPA and Δ OPB

1) ∠OPA ≅ ∠OPB . . . . . . . . . . . ( seg OP ⊥ chord AB)

2) seg OP ≅ seg OP . . . . . . . .(common side)

3) Hypotenuse OA ≅ Hypotenuse OB . . . . . . (Radii of the same circle)

4) ∴ Δ OPA ≅ Δ OPB . . . . . . . . . (Hypotenuse side theorem)

5)  seg PA ≅ seg PB . . . . . . . . . . . . (C.S.C.T)

Tangent theorem

Theorem 2:-  A tangent at any point of a circle is perpendicular to the radius at

the point of contact.

Given:- Line l is a tangent to the circle with centre O at the point of contact A.

To prove: line l ⊥ radius OA.

Tangent Theorem

Tangent Theorem

Proof :- 1) Assume that, line l is not perpendicular to seg OA. Suppose, seg OB is drawn perpendicular to line l.

2) Of course B is not same as A.Now take a point C on line l such that A-B-C and

BA = BC.

3) Now in, Δ OBC and Δ OBA

i) seg BC ≅ seg BA ........ (Construction)

ii) ∠ OBC ≅ ∠ OBA....... (Each right angle)

iii) seg OB ≅ seg OB…….(Common side)

iv) ∴ Δ OBC ≅ Δ OBA.......... (SAS test)

v) ∴ OC = OA…………….. (C.S.C.T)

5) But seg OA is a radius.

 ∴seg OC must also be radius.

 ∴ C lies on the circle.

6) That means line l intersects the circle in two distinct points A and C.

7) But line l is a tangent. ........... (Given)

8) ∴ It intersects the circle in only one point.

9) Our assumption that line l is not perpendicular to radius OA is wrong.

      ∴ Line l ⊥ radius OA.

Tangent segment theorem

Theorem 3: - Tangent segments drawn from an external point to a circle are congruent.

Given:-  seg PD and seg QD are the tangent of the circle

To prove:-   seg DP ≅ seg DQ

Construction:-  Draw radius AP and radius AQ 

Tangent segment theorem

Proof: In Δ PAD and Δ QAD,

i) seg PA ≅ seg AQ…………..( radii of the same circle.)

ii) seg AD ≅ seg AD………….. (Common side)

iii) ∠APD = ∠AQD = 90°…......   (Tangent theorem)

iv) ∴ Δ PAD ≅ Δ QAD……….. (SAS Test)

v) ∴ seg DP ≅ seg DQ………….(C.S.C.T)

Also read Metallurgy