Question

Prove that if chords of congruent circles subtend equal angles at their centre, then the chords are equal.

Answer 1

Hint: We will use the property of congruence of triangles. One of which is given as SAS Congruence rule. This rule is stated as “If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.”

Complete step-by-step answer:
It is given that two chords of congruent circles subtend equal angles at their center.
We have to show that the chords are equal.
Now because the circles are congruent so we can consider the same one circle, because a circle is always congruent to itself.
Analyzing the above situation, we have figure of the question as,

Let us consider two congruent circles (circles of same radius) with centers as O and O.
Let the triangle be \[\Delta AOB\]and \[\Delta COD\]
It is given that \[\angle AOB=\angle COD\]………..(i)
We know that the radius of the congruent circles is equal, which gives,
OA = OC ………..(ii)
Again, we know that radius of the congruent circles are equal, which gives,
OB = OD………(iii)
Now we will use SAS congruence theorem which is stated as,
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
Here we have in \[\Delta AOB\] and \[\Delta COD\] all the three conditions of SAS congruence rule are satisfied which is given by equation (i), equation (ii) and equation (iii).
So, by applying rule of SAS congruence in \[\Delta AOB\] and \[\Delta COD\], we have,
\[\Delta AOB\cong \Delta COD\].
Hence, both the above triangles are congruent.
Now CPCT states that if two or more triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other.
Then by the rule of CPCT the sides are equal, which gives,
AB = CD, which was required to prove.
Hence it is proved that chords of congruent circles subtend equal angles at their centers, then the chords are equal.

Note: The possibility of error or confusion in this question can be at the point where you have to consider to draw the circles. you can get confused in selecting 2 different circles to work with or just one circle. both the methods are correct because the circle is given to be congruent. So, either you select two circles to work with or just one both are correct.