Answer:
(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).
Answer: (1, 4) and (3, 2)
