Reflecting shapes in a mirror line

Reflecting Shapes in a Mirror Line

This is a typical question from a Dulwich College 11+ Maths paper, and it asks you to draw a reflection of the triangle in the mirror line shown on the chart.

Dulwich papers tend to be a bit tricky, and this is not the easiest version of this kind of reflective symmetry question.

For a start, the mirror line is drawn at 45 degrees rather than being horizontal or vertical, and it doesn’t help that the diagram is a bit ‘squashed’, which means the mirror line is actually at around 40 degrees rather than 45!

So how should you do it?

The first thing to do is to imagine that you were looking at yourself in the mirror from, say, 30cm away.

Your reflection will appear ‘in’ the mirror, but it won’t be on the surface of the mirror, will it?

It’ll actually seem to be 30cm ‘behind’ the mirror – which is exactly the same distance as you are in front of it.

That’s important, and you’ll have to use that fact when you do the question.

The basic steps are these:

  1. Plot the ‘vertices’ (or corners) of the reflected shape one by one by drawing a small cross in pencil.
  2. Join them up using a ruler and pencil.

In order to plot each corner, you need to imagine that the corner is your face and that the mirror line is the mirror.

To see your reflection, you have to be standing right in front of the mirror – looking at an angle of 90 degrees to the mirror – so to ‘see’ the reflection of a corner, you have to do the same, looking at an angle of 90 degrees to the mirror line.

The distance from your face to the mirror is the same as the distance to the spot ‘behind’ the mirror where you see your reflection.

In the same way, the distance from the corner to the mirror line is the same as the distance to the spot ‘behind’ the mirror line where the reflected point should go.

If you use the diagram at the top of this article to help you, you should be able to see that the top of the triangle is one-and-a-half diagonal squares away from the mirror line.

That means you need to go another one-and-a-half diagonal squares the other side of the mirror line (continuing in the same direction) in order to plot the reflected point.

Now repeat this for the other corners of the triangle, which are four-and-a-half and three diagonal squares away from the mirror line.

Once you’ve done that, you can join up all three points using a ruler and pencil to make the reflected triangle.

Once you get the hang of it, you may not even need to plot all the corners: if it’s a simple shape like a square or a rectangle, then you might be able to draw it from scratch.

Just make sure you label the shape if the question asks you to.

And that’s it…!

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