Video: Finding the Points Given on a Coordinate System and the Area of the Shape That Results from Connecting Them

Use the graph below to determine the points 𝐴, 𝐵, 𝐶, and 𝐷, and find the area of the shape that results from connecting them.

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Video Transcript

Use the graph below to determine the points 𝐴, 𝐵, 𝐶, and 𝐷 and find the area of the shape that results from connecting them.

So the first thing we’re gonna do in this question is find the coordinates of our points 𝐴, 𝐵, 𝐶, and 𝐷. So what we’re gonna do is take a look at 𝐴 first. Now, if we want to find the coordinates of a point, what we look at first is the 𝑥-coordinate and then we look at the 𝑦-coordinate. And sometimes this is known as going along the corridor and up the stairs. So for 𝐴, we can see that our 𝑥-coordinate is going to be negative four because that’s where 𝐴 lies on the 𝑥-axis. And our 𝑦-coordinate is going to be six. So, that means the coordinate points of 𝐴 are negative four, six. Okay, great! So now we can use the same method to find the coordinates of 𝐵, 𝐶, and 𝐷.

Well, point 𝐵 is gonna be negative four, negative six. Point 𝐶 is gonna be four, negative six. And point 𝐷 is gonna be four, six. So therefore, we’ve solved the first part of the problem because what we’ve done is we’ve determined the coordinates of points 𝐴, 𝐵, 𝐶, and 𝐷. So now what we want to do is find the area of the shape that results from connecting them. So, now what we’ve done is we’ve actually connected the points to make our shape. But what shape is it? And what we have, in fact, is a rectangle, and we know that’s a rectangle because we know that the pairs of sides are, in fact, parallel.

And we know that because if we look at our coordinates of our points, we can see, for example, that 𝐴 and 𝐵 have the same 𝑥-coordinate, 𝐵 and 𝐶 have the same 𝑦-coordinate, 𝐶 and 𝐷 have the same 𝑥-coordinate, and 𝐴 and 𝐷 have the same 𝑦-coordinate. So therefore, we have our rectangle. Well, we know that the area of a rectangle can be found by multiplying the length and the width. Well, first of all, we can see that the width of our rectangle is eight units, and we can get that by counting the squares along.

Or we could have also found the width of our rectangle by finding the difference between the 𝑥-coordinates of 𝐴 and 𝐷. So we’d have four minus negative four. I also here put the modulus or absolute value sign around because what we’re looking at is the magnitude because if we’re trying to find length, we’re only interested in positive value. And this would give us eight because four minus negative four, if you subtract a negative, it’s the same as adding a positive.

Well, now we can find the length of the rectangle. We could do this again with two ways. One, we could count the squares, and we could see that it’s 12 units long. Or again, we could find the difference in the 𝑦-coordinates, this time, of 𝐷 and 𝐶. Well, what we’d have is negative six, because that’s the 𝑦-coordinate of 𝐶, minus six, which would give us negative 12. But as we already said, we’ve got the modulus or absolute value here because all we’re interested in is the magnitude because we want to find the length of the side. And the modulus of negative 12 is just 12, so the area is gonna be equal to 12 multiplied by eight, which is gonna give us 96 square units.

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