Chords AB and CD are drawn in the circle, intersecting at point K, DK = 8cm, CK = 12cm.

Chords AB and CD are drawn in the circle, intersecting at point K, DK = 8cm, CK = 12cm. Triangle area AKD = 24cm. Find the area of a triangle CBK.

In the triangles AKD and CKВ, the angle DKA = ВKS as the vertical angles at the intersection of the chords AB and СD.
The angle ADK rests on the arc AC, the angle ABC also rests on the arc AC, which means the angle ADK = ABC.
Then the triangles AKD and СКВ are similar in two angles.
The sides of the DC and ВK are similar, then the coefficient of similarity of the triangles is: K = DC / ВC = 8/12 = 2/3.
The ratio of the areas of similar triangles is K ^ 2, then Sacd / Scvk = (2/3) ^ 2 = 4/9 = 24 / Svk.
Ssvk = 24 * 9/4 = 54 cm2.
Answer: The area of the СВK triangle is 54 cm2.