Layla draws a square on this coordinate grid. Three of the vertices are marked. What are the coordinates of the missing vertex?
So the first piece of information that we’re told is that Layla draws a square on the grid. This is important because we know that squares have lines of symmetry and they have four sides that are equal. All this could come in useful later on. The second piece of information we’re told is that three of the vertices are marked.
What are vertices? There is a similar word in the final question where we’re asked to find the coordinates of the missing vertex. Well the vertices of a shape are its corners. And so, three of the corners of the square are marked on the grid: one, two, three. But of course, we know that squares have four corners. And so that’s why our question is asking us to find the coordinates of the missing vertex or missing corner.
One thing we could do to start off with is to use a ruler and to connect the three points that we have already. Drawing on a diagram like this can often be a really good idea. And although we’re not told to draw the square for ourselves, actually seeing the square can help us to find those missing coordinates.
At the moment, we’ve drawn two diagonal Lines how does this look like a part of a square. Try turning a head to look at the grid from an angle. Can you see where the fourth corner is likely to be? If you turned your head, you should have seen something like this.
And so we can predict that the missing vertex is going to be in this part of the grid here. So we can see that although it’s at an angle, it is still a square. One thing we know about squares is that they’re symmetrical. And so we can imagine a mirror line going down one of the diagonals of Layla’s square. Everything on the left of the mirror line will eventually look like everything that we can see at the moment on the right of the mirror line.
The missing vertex that we’re looking for is the opposite one to this one here. We need to reflect it across the mirror line. How many squares away from the mirror line is this corner? One, two, three, four squares to the right of the mirror line. So to find the missing corner, we need to count four squares in the opposite direction: one — notice that we’re counting from the mirror line and not from the 𝑦-axis. At the moment, we’re not really thinking about the axes; we’re just trying to make sure that our square is symmetrical. Let’s carry on counting squares: two, three, four.
And so, we found our missing vertex. But how would we describe its position? What are the coordinates of our missing vertex? We know that coordinates are written as a pair of numbers inside brackets with a comma in between. The first number is the position on the 𝑥-axis, the number of squares we have to count along. And the second coordinate is the position on the 𝑦-axis or the number of squares that we need to count up.
It’s very important that we write coordinates in the correct order so that we mark the position in the correct place. There are lots of little rhymes and sayings that we can use to help us remember which order to write the numbers in. For example, one is “across the whole way then up the stairs.” So we need to count across first and then up. However you choose to remember it, just make sure that you do.
So we’re going to start from zero. And first of all, we’re going to look at the position on the 𝑥-axis. How many squares across? One, two, three. And we can see that our vertex is level with this. Because we’ve counted to the left, it’s a negative number. And so, our first coordinate is minus three.
Next, we start from zero again and count how many squares up. One and we can see that our point is level with this. We’ve counted one square up. So our second coordinate is the number one. And so we’ve answered the question. The coordinates of the missing vertex are minus three, one.
Of course, there’s one more thing we could do to check our answer and it is always worth doing. And that’s to use a ruler to connect up the vertices just to check that we have plotted correctly and that the last vertex is in the correct place.
One of the things that makes this question tricky is that the square has been drawn at an angle. If it was drawn in a different position to the way that we would usually think of the square, then perhaps the question would have been a little easier to work out. So that’s why tilting our head is a good idea. It helps us to check the shape from a different angle.
And so, we can see that we have plotted it correctly and it does make a square. The coordinates of Layla’s missing vertex are minus three, one.