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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.

02:43

Video Transcript

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

So according to the question, the line of reflection is the 𝑦-axis. So that’s this line here. Now when we’re carrying out a reflection transformation, we draw lines through each point which are perpendicular to the line of reflection. So for example, the line between 𝐢 and 𝐢 dashed, this would be part of that line. And between 𝐡 and 𝐡 dash, this would be part of that line. And for the line between 𝐴 and 𝐴 dashed, which is the whole point of this question, this would be part of that line there.

Now if we extend those lines the same distance the other side of the line of reflection, we can find out where the original points 𝐴, 𝐡, and 𝐢 were. Now 𝐢 dash was four units to the left of the line of reflection. So if we extend that line four units the other side of the line of reflection, we’ll find out where point 𝐢 was.

𝐡 dashed is eight units to the left of the line of reflection. So if we extend the line the same distance in the other direction, we find out that the original point 𝐡 was here.

And lastly, 𝐴 dash is three units to the left of the line of reflection. So going the same distance in the other direction tells us the original point 𝐴 was here.

So the line from 𝐴 to 𝐴 dash is perpendicular to the line of reflection, and the line of reflection exactly cuts in half. So this distance is the same as this distance. Likewise, for the line between points 𝐡 and 𝐡 dashed, this distance is the same as this distance. And again, for 𝐢, this distance is the same as this distance.

So our original shape 𝐴𝐡𝐢 would’ve looked like that. Now the point to note here is, if I take shape 𝐴𝐡𝐢 and reflect it in the 𝑦-axis, I get shape 𝐴 dash 𝐡 dash 𝐢 dashed. That’s the inverse reflection of going from 𝐴 dash 𝐡 dash 𝐢 dash to 𝐴𝐡𝐢.

So we can see that the original transformation, the original reflection, was in this direction. So to work out where the original shape was from the transformed shape, we can just do the inverse reflection back to the original position.

Okay. All very interesting, but we haven’t actually answered the question yet. It said to determine the original coordinates of point 𝐴. So here’s point 𝐴, and it’s got an π‘₯-coordinate of three and a 𝑦-coordinate of five.

So the original coordinates of point 𝐴 were three, five.