Question

if angle subtended by the chord of a circle at the centre are equal then prove that chords are equal

Answer 1

$PROOF$

We are given two equal chords AB and CD of a circle with centre O as given in the figure.

We want to prove that angle ABC is equal to angle COD

In triangles ABC and triangle COD,

OA = OC. (Radii of a circle)

OB = OD. (Radii of a circle)

AB = CD. (Given)

Therefore,. ∆AOB ≈ ∆COD. (SSS rule)

This gives Angle AOB = Angle COD

(Corresponding parts of congruent Triangle)

HENCE PROVED

Answer 2

Given :

• ∠AOB = ∠COD

To Prove :

• AB = CD

Solution :

Let a circle with centre O and AB & CD are two chords .

In Δ AOB and Δ COD :

$\longmapsto\tt{OA=OD\:(Radius)}$

$\longmapsto\tt{\angle{AOB}=\angle{COD}\:(Given)}$

$\longmapsto\tt{OB=OC\:(Radius)}$

So , By SAS Rule Δ AOB ≅ Δ COD .

Now ,

$\longmapsto\tt{AB=CD\:(By\:CPCT)}$

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Chord :

• The line joining any two points on a circle is called Chord .

Circle :

• The collection of all the points in a plane which are at fixed distance from a fixed point in the plane is called Circle .

SAS Rule :

• Two Triangles are congruent if two sides and the included angle of One triangle is equal to the sides and included angle of other triangle .

• CPCT stands for Corresponding Parts of Congruent Triangle .

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