Question

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

Answer 1

Hint: In this question we may use the fact that the radii of congruent circles are equal. Then we can use the Side Angle Side (SAS) congruence criterion to show that the triangles formed by the radii and chords in the circles are congruent. Thereafter, as Corresponding Parts of Congruent Triangles (CPCT) are equal it can be proved that the chords are also equal.
Complete step-by-step answer:
In this question, we are asked to prove that, if chords of congruent circles subtend equal angles at the center, then the chords are equal. We know that two circles are congruent if their radii are equal.
Here, we are going to use the Side Angle Side (SAS) congruence criterion which states that if in two triangles, any two sides along with the enclosed angle by then are respectively equal then the two triangles are congruent………………………………….(SAS Congruence Criterion)
The process of proving is as follows:
Let us first construct two congruent circles with chords subtending equal angles at the center,

The two circles are drawn with centers A and O and chords BC and PQ. Also, $\mathrm{\angle }BAC=\mathrm{\angle }POQ$.
Now, as we know congruent circles have equal radii.
So, BC=PQ………... (1.1)
Now, in $\mathrm{\Delta }ABC$ and $\mathrm{\Delta }OPQ$,
$$AB=OP$$ (From 1.1)
$\mathrm{\angle }BAC=\mathrm{\angle }POQ$ (Given)
$AC=OQ$ (From 1.1)
So, $\mathrm{\Delta }ABC\cong \mathrm{\Delta }OPQ$ (SAS congruence criterion)
$\Rightarrow BC=PQ$ (CPCT)
Hence, we have got both the chords subtending equal angles at the centers of the two congruent circles as equal.
Therefore, we have proved that if chords of congruent circles subtend equal angles at the center, then the chords are equal.

Note: In this case, we used the SAS congruence criterion. However, one can use any other method of proving the congruence to show that $\mathrm{\Delta }ABC\cong \mathrm{\Delta }OPQ$ it is sufficient to prove that the chords are equal as the sides of congruent triangles should be of same length.