Reflections in a line and in a point

What is it about?

In a reflection in a line $s$, every point $P$ is mapped to an image point $P'$. The following holds:

• $P$ and $P'$ lie on a perpendicular to $s$ (the perpendicular from $P$ to $s$),
• $P$ and $P'$ are the same distance from $s$ – the line bisects the segment $PP'$.

In a reflection in a point $Z$, $Z$ is the midpoint of every segment $PP'$:

A reflection in a point is the same as a half turn ($180^\circ$) about $Z$.

Both reflections are congruence transformations: image and original are congruent, lengths and angles stay the same. Differences:

The orientation tells you in which direction you go round the corners $A \to B \to C$: anticlockwise or clockwise. A fixed line is mapped onto itself (its points may move along it).

A figure has line symmetry if a reflection in a line maps it onto itself, and point symmetry if a reflection in a point (half turn) maps it onto itself.

Try it

1. Drag the corners of the triangle and the points of the line $s$. How can you see that $A$ and $A'$ are the same distance from $s$? What happens if a corner lies exactly on $s$?
2. Press “Fold over”: the triangle is folded over the line like a sheet of paper. Compare the circular arrows in the original and the image.
3. Reflect the flag in a slanted axis. Which way does the reflected flag point?
4. Switch to reflection in a point and press “Turn by 180°”. Does the orientation stay the same? Compare the positions of $AB$ and $A'B'$.
5. Switch on fixed points and fixed lines. Which lines are mapped onto themselves?
6. Drag the image point $A'$: where must $A$ be then?
7. In the mode find the symmetry, find all lines of symmetry and centres of symmetry of the figures.

Tasks

Task 1: Construct the image

Load task In your exercise book, construct the image of the triangle under the reflection in $s$. Describe the steps. Then check with “Fold over”.

Task 2: Reflection in a point in your head

Load task $Z$ is 6 squares to the right of $A$ and 4 squares above $A$. Where is $A'$?

Task 3: Find the line

A triangle and its image under a reflection in a line are given, but the line is missing. How do you find it?

Task 4: Symmetry of letters

Which of the letters H, T, E, N, F have line symmetry, which have point symmetry? Check in the mode find the symmetry.

Task 5: The parallelogram

Lena says: “The diagonals of a parallelogram are lines of symmetry.” Check her claim. Load figure

Task 6: Reflecting twice

Reflect a triangle twice in the same line. What do you get? And what happens to the orientation if you reflect in two different lines one after the other?

Ideas for the classroom

• Use: About grade 7, figures with line and point symmetry: properties of reflections in a line and in a point, constructing image points with a set square, recognising symmetry. Suitable for introducing the topic (folding over as a hands-on idea) and for practice.
• Controls: Corners, the points of the line, the line itself, the centre and the whole figure can be dragged; image points too. Switch off “Show the image” for construction tasks in exercise books. In “find the symmetry”, students draw lines by dragging and tap a suspected centre; wrong attempts show the non-matching image in red.
• Model: All points lie on grid points of the squared paper (one square $= 0.5\,\text{cm}$); with a slanted line, image points in general do not lie on grid points. If part of the image lies outside the paper, a note appears. If $P$ and $Q$ coincide via a slider or link, the point moved last moves on by one square, so there is always a line. If the corners lie on one line, the simulation says that this is not a proper figure.
• Common stumbling blocks:
  • Reflecting “horizontally” instead of perpendicular to the (slanted) line.
  • In a reflection in a point, students only “turn round” without marking the same distance beyond $Z$.
  • “The diagonal of a parallelogram is a line of symmetry.”
  • “A reflection in a point reverses the orientation.” – It is a half turn ($180^\circ$) and keeps it.