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Video: Finding the Original Coordinates of the Vertices of a Triangle given Its Vertices After Reflection

Tim Burnham

The given figure shows triangle 𝐴′𝐵′𝐶′ after a reflection over the 𝑦-axis. Determine the original coordinates of point 𝐴.

02:46

Video Transcript

The given figure shows triangle 𝐴 dash 𝐵 dash 𝐶 dash after a reflection over the 𝑦-axis. Determine the original coordinates of point 𝐴.

So according to this question, the line of reflection is the 𝑦-axis. So according to the question, the line of reflection is the 𝑦-axis. So that’s this line that I’ve marked in a dotted orange line. When we’re carrying out a reflection transformation, we draw lines which are perpendicular to the line of reflection through each point. So for example for point 𝐶, we’d be drawing this line here which is at ninety degrees to the line of reflection of a point 𝐶. We’ll be doing this line here. And for point 𝐴, which of course is our point of this question, that would be our line. Then we can extend those lines the same distance the other side of the line of reflection.

So for example, 𝐶 dash is four units away from the line of reflection, so we need to extend that line four units in the other direction to find the original point 𝐶. For 𝐵, we were eight units to the left of the line of reflection, so I’ve got to go eight units to the right of the line of reflection to find the original point 𝐵. And likewise for 𝐴 dash, that’s three points to three units to the left of the line of reflection. So to find the original point 𝐴, we extend the line three units to the right of the line of reflection in this case. So this distance is the same as this distance, this distance here is the same as this distance here, and this distance here is the same as this distance here. So our original shape 𝐴𝐵𝐶 would’ve looked like that.

Now the point to note at this stage is to find reflect 𝐴𝐵𝐶 in the 𝑦-axis, I get 𝐴 dash 𝐵 dash 𝐶 dash, that’s the inverse, the opposite if you like, of reflecting 𝐴 dash 𝐵 dash 𝐶 dash in the 𝑦-axis, which would take it back to triangle 𝐴𝐵𝐶. And this inverse relationship means that the original transformation was in this direction. So in order to work out what the original triangle was, I can just do the inverse reflection going back from 𝐴 dash 𝐵 dash 𝐶 dash to 𝐴𝐵𝐶 to find the original points. So if we look at point 𝐴, it’s got an 𝑥-coordinate of three and a 𝑦-coordinate of five. So the original coordinates of point 𝐴 are three, five.