Video: Finding the Perimeter and Area of a Rectangle given the Coordinates of Its Vertices to Solve a Word Problem

In a house design that uses a coordinate plane, the vertices of a living room are plotted at (2, 2), (2, 8), (9, 8), and (9, 2), where the coordinates are measured in meters. Determine the perimeter and the area of the room.

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Video Transcript

In a house design that uses a coordinate plane, the vertices of a living room are plotted at two, two; two, eight; nine, eight; and nine, two, where the coordinates are measured in meters. Determine the perimeter and the area of the room.

So the first thing we’ve done here is drawn a sketch of the scenario. So we have the four vertices plotted. And then we can join these up to make our living room. So now what we want to do is find the lengths of our sides. So we can see that our living room has horizontal and vertical sides. We know that because, for instance, if we look at the top line, if we look at the vertices and their coordinates at either end, we’ve got the same 𝑦-coordinate. So therefore, this is going to be a horizontal line. If we look to the far right, then we’ve got the same 𝑥-coordinate. So this is going to be a vertical line.

So if we want to work out the length of the top line, what we’re gonna do is find the difference between the 𝑥-coordinates. So we have nine minus two, which is gonna be equal to seven. If we hadn’t drawn the diagram, we could still see this because if we look at our points that we’ve been given, we could see that the two that have the same 𝑦-coordinate here are two, eight; nine, eight. So therefore, we find the difference between their 𝑥-coordinates, again, which would give us seven. And we know that the units are in meters.

Now, what we’d expect by looking at the diagram is that the distance between our point two, two and nine, two would be the same because it does look like we have a rectangle. Let’s double-check this. Well, yes, once again, if we look at the 𝑥-coordinates and the difference between them, we have nine minus two, so it’s gonna give us seven. So we know that’s seven meters long as well.

So now let’s take a look at our vertical sides. Well, for our vertical sides, what we’re looking for are the pairs of coordinates who have the same 𝑥-coordinates. So we’ve got nine, eight; nine, two and two, eight; two, two. And then what we want to do is find the difference between their 𝑦-coordinates. And here it’s gonna be eight minus two for both, which is gonna give us six. So therefore, we know the height is six meters.

And as we’ve got two pairs of equal parallel sides, we know that what we’re looking at here is a rectangle. So now what we can do is find the perimeter and the area. Well, the perimeter is the distance around the outside, so it can be seven add six add seven add six, which is gonna give us an answer of 26 meters. But then if you want to find the area, the area is equal to the length multiplied by the width. So it’s gonna be equal to seven multiplied by six, which is gonna give us an area of 42 meters squared.

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