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This problem trains for: SAT-I, AMC-8, GMAT.
A rectangle ABCD with area 34 is inscribed in a circle. With respect to the circle, the vertices ABC are kept fixed and the vertex D is moved along the circle to position D' where D'C=BD'. What is the area of the quadrilateral ABCD'?
Denote the lengths of the sides of the rectangle with a and b.
The diagonal AC splits the rectangle in two right triangles. Each of them has an area that is half of the area of the rectangle:
Since the point D' is on the circle and is subtended by the diameter AC it follows that the angle AD'C=90° (by the theorem of Thales).
Therefore, since D'C=a and the diameter AC is unchanged, triangles ADC and AD'C are congruent (SAS) and D'A=b. Since they are congruent, they have the same area.
The area of ABCD' is the same as the area of ABCD.