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

it's easy.

diagonals of rectangle bisect each other.
so, mid-pt (3,2).
so,diagonal pass through it.
=> c=-4.

Now,1st method:
let's assume one of the point to be (x,y) as one of the remaining vertex on the given diagonal.
rotate the line by 45 degree clockwise,and it'll pass through one of the given vertex.It gives you one eqn of line.
Now, rotate it 45 anti-clk it'll pass through other point. hence, 2nd eqn.
find point of intersection of both lines, u'll get one of the vertex.
u have mid point so 4th vertex could be found.

2nd method (easy one):
assume co-ord (a,b) and (c,d)
u know their mid pt as diag. bisect.
also, u can make sides of rectangle equal for more conditions and also these points satisfy the given equation of line,i.e., diagonal.
u'll get required values.

diagnols of a rectangle do not make an angle of 45degree with the sides

so we should not use the rotation technique . rather we should first find the mid point of the given point ie(3,2)

. because the slope of line is 2

ie. tanA=2(say A be the angle )

find the sine and cosine of the angle and then apply the distance from the mid point to find the points. distance from the mid point can be found as the distance of (3,2) from (5,1) as the diagnols are equal.