10.2
1. Recall that two circles are congruent if they have the same radii. Prove that equal
chords of congruent circles subtend equal angles at their centres.
Gu
Answers
Answer:
Q.Two circles are congruent if they have the same radii. Prove that equal
wo circles are congruent if they have the same radii. Prove that equalchords of congruent circles subtend equal angles at their centres.
Ans.
As per the theorem; equal chords (of a circle) subtend equal angles at the centre. Hence, it is clear that equal chords of congruent circles would subtend equal angles at their centres.
Draw chords
Q
R
and
Y
Z
in
2
congruent circles respectively. Join the radii
P
R
,
P
Q
and
X
Y
,
X
Z
respectively.
Given that chords subtend equal angles at centre. So
∠
Q
P
R
=
∠
Y
X
Z
.
We need to prove that chords are equal. i.e
Q
R
=
Y
Z
.
Since the circles are congruent, their radii will be equal.
P
R
=
P
Q
=
X
Z
=
X
Y
Consider the
2
triangles
Δ
P
Q
R
and
Δ
X
Y
Z
.
P
Q
=
X
Y
(
Radii
are equal
)
∠
Q
P
R
=
∠
Y
X
Z
⎛
⎜
⎜
⎜
⎝
Chords
subtend
equal angles
at centre
⎞
⎟
⎟
⎟
⎠
P
R
=
X
Z
(
Radii
are equal
)
By SAS criteria
Δ
P
Q
R
is congruent to
Δ
X
Y
Z
.
So by CPCT (Corresponding parts of congruent triangles)
Q
R
=
Y
Z
.
Hence proved if chords of congruent circles subtend equal angles at their centres then the chords are equal
Mark my answer BRAINLIEST if it helps.