Video: Determining the Pair of Triangles That Represents a Reflection in the ๐‘ฅ-Axis

Which pair of triangles represents a reflection in the ๐‘ฅ-axis?

02:24

Video Transcript

Which pair of triangles represents a reflection in the ๐‘ฅ-axis?

The ๐‘ฅ-axis is this horizontal line. We could call that the line with equation ๐‘ฆ equal zero. Now, weโ€™re looking for the pair of triangles which represent a reflection in this line. Now, when we reflect a shape, we flip it. The two triangles will have the same size, and theyโ€™ll be located the same distance from the ๐‘ฅ-axis but on opposite sides. So, letโ€™s look at some of these pairs.

Weโ€™ll begin by looking at shape ๐ด and ๐ต. For shape ๐ด and ๐ต, this first vertex is two units away from our mirror line. For both shapes, this second vertex is five units away from the mirror line on opposite sides. And our third vertices are both three units away from the mirror line on opposite sides. We see that each of the points are the same distance from the ๐‘ฅ-axis on opposite sides. And the shape is flipped but otherwise unchanged. That is indeed a reflection in the ๐‘ฅ-axis. So, thatโ€™s a good indication of us that the pair of triangles that represent the relevant reflection are ๐ด and ๐ต. But letโ€™s check and see whatโ€™s happened between the other pairs.

Letโ€™s look at shape ๐ด and ๐ถ. Once again, comparing the relevant vertices of our shapes, we see that they are the same distance from the ๐‘ฆ-axis on opposite sides. And the shape is flipped over the ๐‘ฆ-axis but otherwise unchanged. In this case then, shapes ๐ด and ๐ถ represent a reflection in the ๐‘ฆ-axis.

Now, what about shapes ๐ต and ๐ถ? Well, there are two ways to describe this. We add in a diagonal line with the equation ๐‘ฆ equals negative ๐‘ฅ. Now, we compare the vertices of ๐ต and ๐ถ. This time, our vertices are the same distance from the line ๐‘ฆ equals negative ๐‘ฅ but on opposite sides. Otherwise, shapes ๐ต and ๐ถ, apart from being flipped โ€” remember, thatโ€™s a reflection โ€” remain the same shape and size. So, we could say that shapes ๐ต and ๐ถ represent a reflection in the line ๐‘ฆ equals negative ๐‘ฅ.

If we look carefully though, we can even say the shape has been rotated 180 degrees about the origin. Thatโ€™s the point zero, zero. In this case though, the pair of triangles we were looking for were ๐ด and ๐ต.

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