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Video: Finding the Coordinates of the Vertices of a Quadrilateral after Reflection

Tim Burnham

What is the image of 𝐴𝐵𝐶𝐷 under the rotation (𝑥, 𝑦) → (−𝑦, 𝑥)?

03:31

Video Transcript

What is the image of 𝐴𝐵𝐶𝐷 under the rotation 𝑥, 𝑦 maps to negative 𝑦, 𝑥?

Now we’ve been given a figure over here which shows shape 𝐴𝐵𝐶𝐷. Now the coordinates of point 𝐴 are four, three.

So the 𝑥-coordinate is four and the 𝑦-coordinate is three. The coordinates of point 𝐵 are five comma three. So the 𝑥-coordinate is five and the 𝑦-coordinate is three. Point 𝐶 is at five, four with an 𝑥-coordinate of five and a 𝑦-coordinate of four. And point 𝐷 has an 𝑥-coordinate of four and a 𝑦-coordinate of five.

Now under this transformation, what was the 𝑥-coordinate on the original shape becomes the 𝑦-coordinate in the transformed shape. And the negative of what was the 𝑦-coordinate on the original shape becomes the 𝑥-coordinate on the transformed shape. So if we just write down the coordinates of the four Points 𝐴, 𝐵, 𝐶, and 𝐷 and then we’ll apply the transformation to see where they moved to.

Well as we said, what was the 𝑥-coordinate becomes the new 𝑦-coordinate. So, for point 𝐴, the 𝑦-coordinate will become four. And let’s call the transformed point 𝐴 dash. For 𝐵, what was the 𝑥-coordinate was five will become the 𝑦-coordinate, and we’ll call the transformed point 𝐵 dashed. The old 𝑥-coordinate from the original 𝐶 point becomes the 𝑦-coordinate in the transformed point 𝐶 dashed, and the original 𝑥-coordinate for the 𝐷 point becomes the 𝑦-coordinate for the transformed point 𝐷 dashed.

And now we need to transform what was the original 𝑦-coordinate by taking the negative value and installing that as the 𝑥-coordinate of the transformed point. So for point 𝐴, three gets transformed to negative three; for point 𝐵, the 𝑦-coordinate of three gets transformed to negative three; for point 𝐶, the 𝑦-coordinate of four gets transformed to negative four; and for point 𝐷, the 𝑦-coordinate of five gets transformed to the negative of that, negative five, as the 𝑥-coordinate on the transformed point 𝐷 dashed. So we can write down the coordinates of the transformed points 𝐴 dashed, 𝐵 dashed, 𝐶 dashed, and 𝐷 dashed in a little answer box there.

Now just before we go, let’s plot those points on our axes over here to see what the transformation looks like. So, 𝐴 dashed is gonna be a negative three, four, so that’s up here; 𝐵 dashed is gonna be a negative three, five, so that’s up here; 𝐶 dashed is negative four, five, so that’s up here; and 𝐷 dashed is negative five, four, so that’s up here.

And joining those points up in the correct order shows us what the transformed figure would look like; it’s a rotation. In fact, it’s a rotation of ninety degrees in the counterclockwise direction about the origin, but since the question didn’t ask that, we don’t have to give that.