The points (1,3) and (5,1) are two opposite vertices of a rectangle. The other two vertices are lie on the line y =2 x+ c where c is the constant, then co-ordinates of other two vertices are

Step-by-step solution

Given the rectangle with opposite vertices at A = (1, 3) and C = (5, 1) , the other two vertices lie on the line y = 2x + c . The diagonal AC has midpoint M = ( 1+5 2 , 3+1 2 ) = (3, 2) . Since M lies on the line, 2 = 2(3) + c implies c = -4 , so the line is y = 2x - 4 . The diagonal length is AC = (5-1)^2 + (1-3)^2 = 20 = 2 5 . The distance from M to any vertex is half this, 5 . Let B = (x, 2x - 4) . The squared distance MB^2 = (x - 3)^2 + (2x - 6)^2 = 5 . Simplifying, (x - 3)^2 + 4(x - 3)^2 = 5 , so 5(x - 3)^2 =