A rectangle is inscribed in a circle. The coordinates of the endpoints of one of its sides are (1, 2) and (5, 5) . If the equation of a diameter of the circle is x + 2y - 5 = 0 , then the area of the rectangle is _____.

Step-by-step solution

Let the given endpoints of the side be A(1, 2) and B(5, 5) . The length of the side AB = (5 - 1)^2 + (5 - 2)^2 = 16 + 9 = 5 . The midpoint of AB is M ( 1+5 2 , 2+5 2 ) = (3, 7 2 ) . The slope of the line AB is m = 5 - 2 5 - 1 = 3 4 . The perpendicular bisector of AB passes through the center of the circle. Its slope is - 4 3 . Equation of the perpendicular bisector: y - 7 2 = - 4 3 (x - 3) 8x + 6y - 45 = 0 . The center of the circle lies on the intersection of the perpendicular bisector and the given diameter x + 2