𝐴𝐵𝐶𝐷 is a parallelogram. 𝐴 is the point negative one, one. 𝐵 is the point four, one. And 𝐷 is the point negative three, negative three. Part a), draw 𝐴𝐵𝐶𝐷 on the centimetre grid below. Part b), work out the area of 𝐴𝐵𝐶𝐷.
So, the first thing we need to do in this question to answer part a is plot the three points that we’ve been given. So, I’m gonna plot point 𝐴, which is negative one, one, remembering that the first number is the 𝑥-coordinate and the second number is the 𝑦-coordinate. But we can also think of it is along the corridor and up the stairs.
So, I’ve plotted point 𝐴. So, it’s negative one on the 𝑥-axis and one on the 𝑦-axis. Now, I need to plot point 𝐵, which is four, one. And as you can see for point 𝐵, we’ve gone along the 𝑥-axis four and up the 𝑦-axis one. So, that’s point 𝐵 plotted. And then, we have point 𝐷. And Point 𝐷 is at negative three, negative three. So, I’ve plotted that on the graph as well. So, now, how do we plot point 𝐶? Well, we need to find out what this is.
Well to help us, what I’m gonna do is join the other points first. So, now, to help us plot point 𝐶, we have to think of a couple of properties of a parallelogram. First of all, opposite sides are parallel. And also, opposite sides are of equal length. So therefore, we can see that 𝐴𝐵 is five units long. So therefore, 𝐷𝐶 must also be five units long. But also, 𝐴𝐵 must be parallel to 𝐷𝐶. So therefore, if 𝐴𝐵 is a horizontal line, 𝐷𝐶 must also be a horizontal line. So, I’ve drawn our horizontal line 𝐷𝐶. It’s five units long and its horizontal. So, now, I can mark our point 𝐶.
So, now, I’ve completed parallelogram 𝐴𝐵𝐶𝐷, and drawn it on our centimetre grid below. And we can see that it’s correct because we’ve got two pairs of parallel sides. The sides are also equal length when they are opposite. Okay, now, let’s move on to part b.
In part b, what we need to do is work out the area of 𝐴𝐵𝐶𝐷. So, to work out the area of 𝐴𝐵𝐶𝐷, we need to use this formula we know. And that is that the area of a parallelogram is equal to the base multiplied by the perpendicular height. And just reminding ourselves, perpendicular means at right angles to. And the reason we say this is because it is not the slope height. So, for instance, it is not one of our sloped sides.
And often in parallelogram questions, they give you the length of this side as a bit of a red herring because people will make the mistake of using it when calculating the area. Because, as we said, the perpendicular height is the height for our parallelogram that is at right angles to either one of the horizontal edges. So, we’ve got that there.
And we can see that the perpendicular height is four units. And we’ve also got the base length because we worked that out earlier. That is five units. So therefore, the area is gonna be equal to five, which is our base length, multiplied by four, our perpendicular height, which is gonna give us 20. And we’re told that it’s on a centimetre grid. So therefore, we can add units. And we can say that the area of 𝐴𝐵𝐶𝐷 is 20 centimetres squared.
And just to explain why we can use this formula, what I’ve done is I’ve drawn some orange dashed lines here. Because this shows what would happen if we straightened up our parallelogram. Because if we straightened up our parallelogram, it would turn into a rectangle. And a rectangle would have a height of four and a length of five. We know for a rectangle you just multiply length by height, or length by width. So therefore, that’s why we can use that formula to work out the area of a parallelogram.