Two triangles have been drawn on the coordinate axes. Triangle 𝐵 is a reflection of triangle 𝐴 in the 𝑥-axis. Two of the vertices of triangle 𝐵 are 10, minus 10 and 20, minus 30. What are the coordinates of the third vertex of triangle 𝐵?
This question is quite wordy. So let’s go through it really slowly and think about what each part is telling us. Firstly, we’re told that two triangles have been drawn on the coordinate axes. The coordinate axes are this horizontal line here, the 𝑥-axis, and this vertical line here, the 𝑦-axis. And the two triangles that have been drawn on these axes, we can see, are triangle 𝐴 and triangle 𝐵.
Now, the problem goes on to ask us for some coordinates for part of triangle 𝐵. But we may have noticed the problem with this. Why is this question tricky? Well, our coordinate grid doesn’t have any grid lines. There are no squares for us to look at. And there are no numbers labelled on the axes. How can we write down coordinates if we have no squares and no numbers on the axes?
Well, thankfully, we’re given some facts to help us. Firstly, we’re given the coordinates of the vertices or the corners of triangle 𝐴. These are 10, 10; 20, 30; and the one at the top here which is minus 10, 40. As well as the coordinates of the three vertices of triangle 𝐴, we’re also given these two facts about triangle 𝐵.
Triangle 𝐵 is a reflection of triangle 𝐴 in the 𝑥-axis. In other words, we can think of the 𝑥-axis as a mirror line. Triangle 𝐴 was drawn first. And then each point was flipped across the mirror line, or, in other words, reflected across the 𝑥-axis to give us the three vertices that make up triangle 𝐵. And so this information helps us know where triangle 𝐵 has come from. It’s a reflection of triangle 𝐴 in the 𝑥-axis.
We know that every triangle has three vertices or three corners. And we’re told the coordinates of two of the vertices of triangle 𝐵. One of them is 10, minus 10. The other is 20, minus 30. And of course, the third corner or the third vertex is the one we’re looking for when we’re answering our problem. So the way we can find out the answer to our problem is to use all the coordinates we’ve been given to help and to think carefully about how coordinates behave when a shape is reflected. Particularly, when it’s reflected across an 𝑥-axis.
Let’s start by thinking about point 10, 10. Where does this end up when it’s reflected in the 𝑥-axis? It becomes this point here. Notice that both these vertices are the same distance away from the 𝑦-axis. We can see that they level with each other. This is because when we reflect the coordinate in the 𝑥-axis, the points always stay the same distance from the 𝑦-axis. So the number of squares along or the 𝑥-coordinate or the first number in a pair of coordinates stays the same. And so point 10, 10 reflects to point 10, minus 10. Instead of counting up 10 on the 𝑦-axis, we’ve had to count down 10, which is where the minus 10 part comes from.
So we’ve labelled one of the vertices we were told as 10, minus 10. The second pair of coordinates we’re given is 20, minus 30. Which of the two remaining vertices is found at position 20, minus 30? It’s this one. Let’s go through why. We start at position 20, 30 which is 20 squares across and 30 squares up. We reflect in the 𝑥-axis. And we know that any point reflected in the 𝑥-axis stays the same distance away from the 𝑦-axis. So it’s still going to be 20 squares across.
As we noticed before, the two points are vertically level with each other. And then instead of 30 squares up, the point is moved 30 squares down. So instead of a 𝑦-coordinate of 30, it’s now minus 30. And we’re starting to see how coordinates behave when they are reflected in the 𝑥-axis. Point 10, 10 became point 10, minus 10. Point 20, 30 became point 20, minus 30. Can we see a pattern here that will help us find the coordinates of the final vertex of triangle 𝐵?
Yes. Firstly, we know that the 𝑥-coordinate will stay the same. Both points are 10 to the left of the 𝑦-axis or minus 10. But instead of travelling 40 squares up on the 𝑦-axis, we’re travelling 40 squares down. And so the 𝑦-coordinate instead of being 40 will be minus 40. And so the coordinates of the third vertex of triangle 𝐵 are minus 10, minus 40.
We worked out the answer by thinking about the fact that triangle 𝐵 was a reflection. We then used the coordinates that we were told to label two of the vertices that we knew already. This helped us understand how coordinates behave when they’re reflected in the 𝑥-axis. And by writing the old and the new coordinates next to each other, we could also see a visual pattern in the numbers. The 𝑥-coordinate stayed the same. But the 𝑦-coordinate went from a positive value to the same value but negative. So even though there weren’t any squares and there were no labels on our axes, we had enough information to help us to find the missing coordinate.