If Angles subtended by the chords at the center of circle are equal, then chords are also equal at Algebra Den

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Home >> Circle >> Properties of Circle >> If Angles subtended by the chords at the center of circle are equal, then chords are also equal >>

If Angles subtended by the chords at the center of circle are equal, then chords are also equal

Equal chords subtend equal angles at the center of a circle If Angles subtended by the chords at the center of circle are equal, then chords are also equal Perpendicular from the center of a circle to a chord bisects the chord Line drawn from the center of circle to bisect a chord, is perpendicular to the chord Equal chords are equidistant from the center of circle
Chords equidistant from the center of circle are equal in length

Before you understand property, if Angles subtended by the chords at the center of circle are equal, then chords are also equal, you are advised to read:

What are Chords of Circle ?
What is center of Circle ?

Observe the following diagram:

In the above diagram, we have:
A circle with center O
PQ and RS are the chords of Circle
Angle subtended by chord PQ at the center of circle is ∠ POQ
Angle subtended by chord RS at the center of circle is ∠ ROS

Now, you can observe also that:
∠ POQ = ∠ ROS = 30° each

So, as per the property of circle "If Angles subtended by the chords at the center of circle are equal then chords are also equal", we get:
Chords PQ = Chord RS

Now, let's prove this property, if Angles subtended by the chords at the center of circle are equal, then chords are also equal in the following way:

Before you prove this property of circle, you are advised to read:

What are Congruent Triangles ?
What is SAS rule of Congruency ?
What are the Corresponding Parts of Congruent Triangles ?
What is the Radii of Circle ?

Observe the following:

In the above diagram, we have:
A circle with center O
∠ POQ = ∠ ROS = 30° each

And we need to prove that:
Chord PQ = Chord RS

In the above diagram, we have two Triangles (as highlighted below):
△ PQO = △ RQO

OP = OR (radii of circle are always equal)
OQ = OS (radii of circle are always equal)
∠ POQ = ∠ ROS (30 degree each - Given)
Therefore, on applying SAS Rules of congruency, we get:
∆ PQO ≅ ∆ RQO

Since, we know that corresponding parts of congruent triangles are equal, so we get:
Chord PQ = Chord RS

Hence, this proves property 2 of circle which says If Angles subtended by the chords at the center of circle are equal then chords are also equal