Question

Prove that if the angles subtended by the chords of a circle at the centre are equal, then chords

are equal.​

Answer 1

Given :-

• A circle with centre O
• AB and CD are chords of the circle and subtend equal angles at the centre. ∠AOB = ∠COD

To Prove :-

AB = CD

Solution :-

In ∆ AOB & ∆ DOC

OA = OD (radius)

∠AOB = ∠COD (given)

OB = OC (radius)

∴ ∆ AOB ≅ ∆ COD

∴ AB = CD (cpct)

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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