# The vertices of the quadrilateral ABCD divide the length of the circumscribed circle in the ratio AB: BC:

The vertices of the quadrilateral ABCD divide the length of the circumscribed circle in the ratio AB: BC: CD: CA = 2: 17: 4: 13. Find the area of the quadrilateral if AC = 8 cm, BD = 9 cm.

Let’s define the degree measures of arcs AB, BC, СD, АD.

2 * X + 17 * X + 4 * X + 13 * X = 360.

36 * X = 360.

X = 100.

Arc BC = 17 * 10 = 170. Then the inscribed angle BAC = 170/2 = 85.

Arc AD = 13 * 10 = 130. Then the inscribed angle ABD = 130/2 = 65.

In triangle AOB, angle AOB = 180 – 85 – 65 = 30.

Let’s define the area of the quadrilateral through the diagonals and the angle of their intersection.

Savsd = (1/2) * АС * ВD * Sin30 = 8 * 9 * 1/4 = 18 cm2.

Answer: The area of the quadrangle is 18 cm2.

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