CAT Relation Between Arc and Chord - Practice Questions & MCQ

Property 1: If two arcs of a circle (or of congruent circles) are congruent then the corresponding chords are equal.

Let's prove this.

We have given a circle C(O,r) in which,  .

And we need to prove chord AB = chord CD.

Case I When and  are minor arc.

Join OA, OB, OC and OD.

In triangle AOB and COD, we have.

                           OA = OC         (Radii of a circle)

                          OB = OD          (Radii of a circle)

                    ∠ AOB = ∠ COD          

Therefore,   ∆ AOB ≅ ∆ COD    (by SAS rule)

This gives         AB = CD     (Corresponding parts of congruent triangles)

Case II When and  are major arc.

In this case,  and  are minor arc

Hence, in both cases, we have AB = CD.

Property 2: If two chords of a circle (or of congruent circles) are equal then their corresponding arcs (semicircular, minor or major) are congruent.

Let's prove this

Let given a circle C(O,r) in which chord AB = chord CD.

We  need to prove , where both  and   are either semicircular, minor or major arcs.

Case I When AB and CD are diameters

In this case,   and   are semicircles with the same radii.

So, 

Thus, .

Case II When chord AB = chord CD, where    and   are minor arcs.

In triangle AOB and COD, we have.

                           AB = CD         (given)

                          OA = OC          (Radii of a circle)

                          OB = OD          (Radii of a circle)

Therefore,   ∆ AOB ≅ ∆ COD    (by SAS rule)

This gives  ∠ AOB = ∠ COD     (Corresponding parts of congruent triangles)

Now,

Case II When chord AB = chord CD, where    and   are major arcs.

In this case,  and  are minor arc

Hence, in all the cases .

Property 2: If two chords of a circle (or of congruent circles) are equal then their corresponding arcs (semicircular, minor or major) are congruent.

Let's prove this

Let given a circle C(O,r) in which chord AB = chord CD.

We  need to prove , where both  and   are either semicircular, minor or major arcs.

Case I When AB and CD are diameters

In this case,   and   are semicircles with the same radii.

So, 

Thus, .

Case II When chord AB = chord CD, where    and   are minor arcs.

In triangle AOB and COD, we have.

                           AB = CD         (given)

                          OA = OC          (Radii of a circle)

                          OB = OD          (Radii of a circle)

Therefore,   ∆ AOB ≅ ∆ COD    (by SAS rule)

This gives  ∠ AOB = ∠ COD     (Corresponding parts of congruent triangles)

Now,

Case II When chord AB = chord CD, where    and   are major arcs.

In this case,  and  are minor arc

Hence, in all the cases .