Quadrilateral ABDC, angle BDC = 90, angle CAB = 90, BD = 9, AC = 7, angle ABC = angle DBC. Find the perimeter.

Since, according to the condition, the angle BDC = CAB = 90, then the segment CB is the diameter of the circle, and the triangles ABC and CBD are rectangular.

Let us prove that triangle ABC is equal to triangle CBD.

Both triangles are right-angled, the angle ABC = DBC by the condition, the hypotenuse of the BC in the triangles is common, then the triangle ABC is equal to the triangle CBD in the hypotenuse and acute angle, the third sign of equality of right-angled triangles.

Then AB = BD = 9 cm, ВD = AC = 7 cm.

The perimeter of the quadrilateral ABCD is: P = 2 * (AC + BD) = 2 * (7 + 9) = 32 cm.

Answer: The perimeter of the quadrangle is 32 cm.