Video Transcript
List the coordinates of the image of 𝐴𝐵𝐶𝐷 after reflection in the 𝑥-axis.
In the grid below, we can see that we have the quadrilateral 𝐴𝐵𝐶𝐷, which we are told will be reflected in the 𝑥-axis. So the 𝑥-axis will be the mirror line of reflection.
When we are reflecting a polygon, we can take each point on the polygon in turn and reflect that point in the mirror line. So let’s take point 𝐴, which has the coordinates negative four, six. When we reflect a point, we consider the perpendicular distance from that point to the mirror line. And point 𝐴 is six units away from the 𝑥-axis. So the reflected point is also at a perpendicular distance of six units away from the mirror line but on the opposite side. We can call this image of point 𝐴 the point 𝐴 prime. And we can read the coordinates as negative four, negative six.
Now we can do the same for point 𝐵, which has the coordinates four, seven. Point 𝐵 is at a perpendicular distance of seven units from the 𝑥-axis. And so the image 𝐵 prime is also a perpendicular distance of seven units from the 𝑥-axis but on the opposite side. The coordinates of 𝐵 prime are four, negative seven.
If we take point 𝐶 next, it is closer to the mirror line of the 𝑥-axis, at just two units away. So the image will also be closer to the mirror line. The image 𝐶 prime has the coordinates three, negative two.
Finally, point 𝐷 is closest to the 𝑥-axis, at just one unit away at a perpendicular distance from the mirror line. So we can reflect it in the 𝑥-axis to the image 𝐷 prime with coordinates negative six, negative one.
Joining the new coordinates, we can create the complete image of 𝐴𝐵𝐶𝐷, which we could call 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime. And we can see that these two polygons do indeed look like reflections of one another.
We can also observe that there is a pattern in the coordinates. In each case, the 𝑥-coordinate has stayed the same between the original point and its image. But the 𝑦-coordinates have changed sign. In each case, they started positive, and in the image they were negative. Note that they all started positive because they were in the top part of the grid. If we were reflecting a point in the 𝑥-axis that had a negative 𝑥-coordinate, then it would be positive in the image.
And in fact, there is a general rule which describes this pattern of how coordinates change after this type of reflection. We can say that for a general point 𝑃 with coordinates 𝑥, 𝑦, a reflection in the 𝑥-axis maps point 𝑃 to point 𝑃 prime with coordinates 𝑥, negative 𝑦. And the useful thing about knowing this property is that it means we don’t always have to draw a point to reflect it. By applying this property, we can determine where the image of a point will be after a reflection in the 𝑥-axis. It is simply that the 𝑥-coordinate stays the same and the 𝑦-coordinate changes sign.
And so either by applying this property or by carrying out the reflection on the grid, we have the answers for the coordinates. 𝐴 prime is negative four, negative six. 𝐵 prime is four, negative seven. 𝐶 prime is three, negative two. And 𝐷 prime is negative six, negative one.
