Question

The points (1, 3) and (5, 1) are two opposite vertices of a rectangle. The other two vertices lie on the line y = 2x + c. Find c and the other remaining vertices.

Answer 1

See Fig.  ABCD is the rectangle in which A = (1, 3), C = (5, 1). Points B and D lie on the line y = 7x + c. The diagonals intersect in (3, 2) which lies on the line y = 7x + c. Therefore, 2 = 2(3) + c or c = - 4. That is, the equation of the diagonal BD is

y = 2x - 4   ...(1)

Suppose M = (3, 2) is the midpoint of the diagonals. Therefore,

MD = MB = (1/2) AC = √5 

Let B be (x, 2x - 4). Therefore

Hence, B = (2, 0) and D = (4, 4).