The perpendicular from the centre to the chord bisects the chord. Using the right triangle, half the chord is 12 cm. Therefore, the complete chord is 24 cm.
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The quadrilateral ABCD is cyclic. The opposite angles of a cyclic quadrilateral are supplementary. So, x + 100° = 180° Therefore, x = 80°.
Yes. If ∠AXB = ∠AYB, then A, B, X and Y are concyclic, provided X and Y are on the same side of AB. This follows from Theorem 10.
Yes, provided X and Y lie on the same side of AB. Angles subtended by the same chord at points on the same segment of a circle are equal.
Yes. If X and Y lie on the same side of AB on the circle, then ∠AXB and ∠AYB subtend the same chord AB. Hence, they are equal, so they cannot be different.