Video Transcript
Rectangle 𝐴𝐵𝐶𝐷 is graphed in the coordinate plane with vertices 𝐴: negative five, negative two; 𝐵: six, negative two; and 𝐶: six, three. Find the coordinates of point 𝐷.
So here, we’re given three points 𝐴, 𝐵, and 𝐶. And we’re told that they’re part of a rectangle and we need to find the coordinates of the final vertex 𝐷. So let’s get some grid paper and see if we can plot these three vertices. So here we have 𝐴 at negative five, negative two; 𝐵 at six negative, two; and 𝐶 at six, three. We can even join these three vertices with two lines as shown. Drawing these three points on the coordinate grid should have already given us a good idea of where we think point 𝐷 will be. However, it’s always a good idea to think through this mathematically.
Firstly, we should recall that the very definition of a rectangle is that it’s a quadrilateral with all four interior angles of 90 degrees. This means there’ll be a right angle here at angle 𝐵 and there’ll also be right angles here at angles 𝐴 and 𝐶. It’s a bit easier to demonstrate in this figure as the line 𝐴𝐵 is horizontal and the line 𝐶𝐵 is a vertical line. So let’s say that we drew a vertical line upwards from point 𝐴. We could also continue this horizontal line from vertex 𝐶. The point where they meet must be vertex 𝐷.
We can also check this using another method which involves knowing a key property of rectangles. And that is that opposite sides in a rectangle are congruent or the same length. If we take a look at the line between the two given coordinates 𝐴 and 𝐵, then from point 𝐴 on the 𝑥-axis to zero on the 𝑥-axis, that’d be five units and from zero on the 𝑥-axis to point 𝐵, that’d be six units, which means that the total length of the line must be 11 units long.
This means that the vertex 𝐷 must be 11 units away from 𝐶 on the 𝑥-axis. And a quick check would show us that, indeed, it’s 11 units long. The line between the given points 𝐵 and 𝐶 is five units. And so 𝐷 must also be five units up from point 𝐴. And so we’ve demonstrated that the point 𝐷 must be at the coordinate negative five, three. And so that’s our answer.
