Question

Equal _____ of the congruent circles subtend equal angles at the centers.​

Answer 1

Answer:

Equal chords of the congruent circles subtend equal angles at the centers.

Answer 2

Equal "Chords" of the congruent circle subtend equal angles at the centers.

Chords in a Circle :

• A chord in a circle is any line segment that has its two endpoints on the circumference of the circle.
• Thus, any line joining points on the circumference is a chord of a circle.
• The diameter is also a chord ( longest chord ) as it extends from points on the circle crossing the center.

Congruent Circles: Congruency in circles apply as - Two circles having the same radius length are congruent.

The trigonometric formula of the angle x subtended by a chord of length L in a circle of radius R at the center is :

               $L = 2*R*sin(x/2)$

For 2 circles C1 and C2 that are congruent and have chords of the same length i.e.

                         $R_{1} = R_{2}$

                          $L_{1}=L_{2}$

  $2*R_{1}*sin(x_{1}/2) = 2*R_{2}*sin(x_{2}/2)$

               $sin(x_{1}/2) = sin(x_{2}/2)$

                          $x_{1} =x_{2}$

Hence, the angles subtended by equal chords in congruent circles are equal.

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