A pair of opposite vertices of a square is (1, 2) and (3,4). Find the coordinates of the remaining vertices of the square.

(3, 2) and (1, 4)

Step-by-step explanation:

Plot the two points on a graph.

The other two points are (3, 2) and (1, 4).

To do this with algebra, it takes a few steps.

The diagonals of a square are perpendicular and bisect each other. You are given opposite vertices, so first, find the midpoint of that diagonal.

((1 + 3)/2, (2 + 4)/2) = (2, 3)

The midpoint of the diagonal is (2, 3).

This diagonal has slope 1 and y-intercept 1, so its equation is

y = x + 1

The perpendicular bisector has equation

y = -x + 5

The two vertices we are looking for, lie in a circle whose center is the midpoint of the diagonals, (2, 3), and whose radius is half of the diagonal.

Use Pythagoras to find the diagonal's length.

2^2 + 2^2 = c^2

c^2 = 8

c = sqrt(8) = 2sqrt(2)

Half of the diagonal is sqrt(2). This is the radius if the circle.

The equation of the circle is

(x - 2)^2 + (y - 3)^2 = (sqrt(2))^2

(x - 2)^2 + (y - 3)^2 = 2

The points of intersection of this circle and the second diagonal are the two vertices you are looking for.

System of equations:

(x - 2)^2 + (y - 3)^2 = 2

y = -x + 5

Use substitution and substitute y with -x + 5 in the equation of the circle.

(x - 2)^2 + (-x + 5 - 3)^2 = 2

(x - 2)^2 + (-x + 2)^2 - 2 = 0

x^2 - 4x + 4 + x^2 - 4x + 4 - 2= 0

2x^2 - 8x + 6 = 0

x^2 - 4x + 3 = 0

(x - 3)(x - 1) = 0

x - 3 = 0   or x - 1 = 0

x = 3 or x = 1

Now we find corresponding y values.

y = -x + 5

x = 3

y = -3 + 5 = 2

This gives us (3, 2).

y = -x + 5

x = 1

y = -1 + 5 = 4

This gives us (1, 4).

(1, 4) and (3, 2)