Since OA and OB are radii, triangle AOB is isosceles. Given ∠AOB = 60°, the other two angles are also 60°. Thus, triangle AOB is equilateral, so AB = OA = 12 cm.
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The perpendicular from the centre bisects the chord, forming two right triangles. Applying the Baudhāyana–Pythagoras Theorem gives the half-chord as √(r² − d²). Hence, the chord length is 2√(r² − d²).
Equal perpendicular distances from the centre imply that the two chords are equidistant from the centre. By Theorem 7, chords equidistant from the centre of a circle are equal. Therefore, AB = GF.
The perpendicular distances of the two chords from the centre are found using the Pythagoras Theorem. Since the chords lie on opposite sides of the centre, the required distance is the sum of these distances, which is seven centimetres.
The two triangles have equal radii as corresponding sides and equal chord lengths as their bases. Therefore, all three corresponding sides are equal, so the triangles are congruent by the SSS Congruence Criterion.