In the picture on the right, the angles between the radii and the chords are equal.

Prove that the chords are of the same length.

Question

In the picture on the right, the angles between the radii and the chords are equal.Prove that the chords are of the same length.

Philoid · Accepted Answer

C is the radius of the circle. ED and FG are two chords.

In ΔCDE we have,

EC = CD [radius of the circle]

∴ ∠CDE = ∠CED ……… (1)

In ΔCFG we have,

FC = CG [radius of the circle]

∴ ∠CGF = ∠CFG ……… (2)

In ΔCDE and ΔCFG we have,

∠CED = ∠CFG [given in the problem] ……… (3)

∠CDE = ∠CGF [from (1), (2) and (3)]

CE = CF [radius of the same circle]

∴∆CDE≅∆CFG [AAS congruency]

∴ ED = FG [similar sides of the triangle.

∴ The chords are of same length.