Question

A rectangle has three of its vertices at points `(3,4),` `(5,–4),` and `(–7, –7).` What is the `y` coordinate of the unknown vertex?

Answer 1

The `y`-coordinate of the fourth vertex is `1`.

Explanation:

The given points can be drawn in the `xy`-plane as follows:
graph{((x-3)^2+(y-4)^2-0.05)((x-5)^2+(y+4)^2-0.05)((x+7)^2+(y+7)^2-0.05)=0 [-14.95, 13.52, -8.37, 5.87]}

From here, it is easy to visualize roughly where the fourth vertex will be—somewhere in `"Quadrant II"`.

Since rectangles are composed of two sets of parallel sides, the `y` distance between two adjacent points will be the same as the `y` distance between the other two points.

What does this mean? Well, the point `(3,4)` is 8 up from its neighbour `(5,"-"4)` (since `4-"(-4)"=8`), so the fourth vertex will also be 8 up from `("-7","-7")`. And what is 8 up from -7? That's right: 1.

We can use this to find the `x`-coordinate as well; just use the `x`-coordinates instead of the `y`'s. The remaining vertex is at `("-9",1)`.

Hope this helps!

Answer 2

`(-9, 1`)

Explanation:

Use :

The vector `(x_1, y_1) to (x_2, y_2)` is `< x_2-x_1, y_2-y_1>`.

The vertices are labelled

`A( 3, 4 ), B( 5, -4 ), C(-7. -7) and D( x, y ).`.

Vector `AD = < x-3, y-4>`

= vector `BC=<-12, -3>`.

Solving,

(`x, y) = (-9, 1)`.