Question

If `(4, 9)` and `(2, 3)` are diagonal corners of a square, what are the coordinates of the other corner of the square?

Answer 1

See a solution process below:

Explanation:

First, we can plot these two points:

graph{((x - 4)^2 + (y - 9)^2 - 0.075)((x - 2)^2 + (y - 3)^2 - 0.075)=0 [-5, 25, -5, 10]

The formula to find the mid-point of a line segment give the two end points is:

`M = ((color(red)(x_1) + color(blue)(x_2))/2 , (color(red)(y_1) + color(blue)(y_2))/2)`

Where `M` is the midpoint and the given points are:

`(color(red)(x_1), color(red)(y_1))` and `(color(blue)(x_2), color(blue)(y_2))`

`M = ((color(red)(4) + color(blue)(2))/2 , (color(red)(9) + color(blue)(3))/2) = (6/2, 12/2) = (3, 6)`

We can plot the midpoint as:

graph{((x - 3)^2 + (y - 6)^2 - 0.075)((x - 4)^2 + (y - 9)^2 - 0.075)((x - 2)^2 + (y - 3)^2 - 0.075)=0 [-5, 25, -5, 10]

We can also find the slope of the line segment. The slope can be found by using the formula: `m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))`

Where `m` is the slope and (`color(blue)(x_1, y_1)`) and (`color(red)(x_2, y_2)`) are the two points on the line.

Substituting the values from the points in the problem gives:

`m = (color(red)(3) - color(blue)(9))/(color(red)(2) - color(blue)(4)) = (-6)/-2 = 3`

Because this is a square the other diagonal will be perpendicular and have a slope of the negative inverse of this slope:

`m_p = -1/3`

Because we have a slope (the negative inverse) and a point (the midpoint) we can find an equation for the line and plot this on the graph. The point-slope form of a linear equation is: `(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))`

Where `(color(blue)(x_1), color(blue)(y_1))` is a point on the line and `color(red)(m)` is the slope. Substituting gives:

`(y - color(blue)(6)) = color(red)(-1/3)(x - color(blue)(3))`

graph{((x - 3)^2 + (y - 6)^2 - 0.075)((x - 4)^2 + (y - 9)^2 - 0.075)((x - 2)^2 + (y - 3)^2 - 0.075)(y + (1/3)x -7)=0 [-5, 25, -5, 10]}

To get from the midpoint to the given points we go left or right 1 unit and up or down 3 units.

To get from the midpoint to the other corners of the square we go left or right 3 units and up or down 1 unti:

`(3 + 3, 6 - 1) = (6, 5)`

`(3 - 3, 6 + 1) = (0, 7)`

graph{(x^2 + (y - 7)^2 - 0.075)((x - 6)^2 + (y - 5)^2 - 0.075)((x - 3)^2 + (y - 6)^2 - 0.075)((x - 4)^2 + (y - 9)^2 - 0.075)((x - 2)^2 + (y - 3)^2 - 0.075)(y + (1/3)x -7)=0 [-5, 25, -5, 10]}