### Video Transcript

The given figure shows triangle π΄ dash π΅ dash πΆ dashed after a reflection over the π¦-axis. Determine the original coordinates of point π΄.

So according to the question, the line of reflection is the π¦-axis. So thatβs this line here. Now when weβre 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 πΆ and πΆ dashed, this would be part of that line. And between π΅ and π΅ dash, this would be part of that line. And for the line between π΄ and π΄ 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 π΄, π΅, and πΆ were. Now πΆ 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, weβll find out where point πΆ was.

π΅ 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 π΅ was here.

And lastly, π΄ 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 π΄ was here.

So the line from π΄ to π΄ 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 π΅ and π΅ dashed, this distance is the same as this distance. And again, for πΆ, this distance is the same as this distance.

So our original shape π΄π΅πΆ wouldβve looked like that. Now the point to note here is, if I take shape π΄π΅πΆ and reflect it in the π¦-axis, I get shape π΄ dash π΅ dash πΆ dashed. Thatβs the inverse reflection of going from π΄ dash π΅ dash πΆ dash to π΄π΅πΆ.

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 havenβt actually answered the question yet. It said to determine the original coordinates of point π΄. So hereβs point π΄, and itβs got an π₯-coordinate of three and a π¦-coordinate of five.

So the original coordinates of point π΄ were three, five.