What is a Circle? Proves | CBSE Class 9 Maths Solutions
What is a circle?
- It is the collection of points or locus of a point which moves in a plane in such a way that a distance from a fixed point is constant.
- The fixed point is called the centre.
- The constant distance from the centre is called the radius.
Above figure is a circle with centre O and radius r (it is denoted by C ( O, r)
1 A point P which lies inside or outside the circle can be OP< r, OP> r or OP=r
2 Circular Disc
A collection of all points lying inside or on the circle is called the circular disc or circular region.
3 Concentric circles
Circles having the same centre with different radii.
In the above figure, there are two circles having same center O with different radii r1 and r2.
It refers to the continues piece of a circle.
In the above figure P1, P2, P3, P4, P5, and P6 are all arcs on the circle.
5 Corresponding chords are equal if two arcs of a circle are congruent.
Given – Arc PQ of circle 1 and arc RS of the second circle are such that PQRS
(6) Prove that If two arcs of a circle are congruent, then corresponding chords are equal.
Given: Arc PQ of a Circle C(O,r) and arc RS of another circle C(O′,r) such that PQ≅RS
To Prove: PQ=RS
There are two cases in the given figure.
Case –I Minor Arcs are given- PQ and RS
Given two circles and two triangles
OP=OQ=O’R=O’S=r ( raddi of two circles are equal)
HENCE ΔPOQ≅ΔRO′S & PQ=RS ( SAS) RULE
Case -2 Major arcs are given PQ and RS
QP and SR are minor arcs so arc PQ ≅ arc RS
So, arc QP ≅ arc SR
QP=SR and PQ=RS
Hence, PQ≅RS ⇒PQ=RS
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