The chords AB and CD of the circle with center O are equal, a) Prove that two arcs with ends A

| September 12, 2021 | education

The chords AB and CD of the circle with center O are equal, a) Prove that two arcs with ends A and B, respectively, are equal to two arcs with ends C and D. b) Find arcs with ends C and D if ∠AOB = 112 °.

Consider triangles AOB and COD.

In the triangle OAB OA = OB = R, in the triangle COD, OС = OD = R, then OA = OB = OС = OD.

By condition, AB = CD, then the triangles AOB and COD are equal on three sides, and therefore the angle AOB = COD, and hence the arc AB is equal to the arc CD, which was required to be proved.

Since the central angle AOB = 112, and the triangles AOB and COD are equal, the central angle COD = 112, then the degree measure of the smaller arc CD is 112, and the larger arc CD = 360 – 112 = 248.

Answer: CD arcs are equal to 112 and 248.



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