Point symmetry

Turn a picture half a turn around one point

You know line symmetry: reflecting in a line, like butterfly wings. Today its sister — point symmetry. Instead of a mirror it has one point: the centre of symmetry S.

The rule for every point of the picture: draw a line from the point through centre S and on the other side measure the same distance. Point A jumps to point A′ so that S is exactly in the middle of segment AA′.

The result looks like a picture turned 180° — upside down and mirrored left-right at the same time. A playing-card jack or queen: you turn the card half a turn and you see the same picture. That is point symmetry in action.

Some shapes are point-symmetric with themselves: they have a point inside around which they turn 180° and look the same. A square, a rectangle, a rhombus and the letters S, N, Z. Today you will learn to construct images and to spot the symmetry.

Sofie designs a class logo and wants it to look the same even if someone hangs it upside down by mistake. Tereza: “Then you need point symmetry. Like cards — the jack is the same at the top and at the bottom.” They try letters: O works, S and N and Z too, but E and T do not. Sofie draws a design from the letter S and two dots opposite each other. She turns the paper 180° — the picture is indistinguishable. “Done. And you know what is funny?” Tereza laughs. “If it were line-symmetric like a butterfly, upside down it would not work.”

Constructing an image in point symmetry

The image of point A in centre S:

Step 1: draw a line through points A and S. Step 2: with a compass measure the distance |AS|. Step 3: measure the same distance on the line past centre S. The intersection is A′.

Check: S must be the midpoint of segment AA′, so |AS| = |SA′|.

The image of a segment, a triangle, anything: map vertex by vertex and join the images. Triangle ABC → you map A, B, C → you join A′B′C′. The image is congruent with the original (same lengths, same angles), just turned 180°.

Properties that are handy to know: a segment maps to a parallel segment of the same length. A line through the centre maps to itself. In point symmetry a picture does not flip as in a mirror — it turns.

Point-symmetric shapes: a shape is point-symmetric when there is a point in which it maps to itself. A test in practice: trace the shape on tracing paper, plant a pencil in the supposed centre and turn 180°. Does it cover? It is symmetric. A square, a rectangle, a rhombus, a circle yes; a triangle never (none, not even equilateral!).

The image of point A in centre S, when |AS| = 3 cm

Step 1: I draw a line through A and S and extend it past the centre.

Step 2: I open the compass to |AS| = 3 cm.

Step 3: I plant it in S and measure 3 cm to the opposite side from A. I mark the intersection with the line A′.

Check: |AS| = |SA′| = 3 cm, so |AA′| = 6 cm and S is exactly in the middle.

The whole trick of point symmetry is in this one step — everything else is just repeating it for more points.

The image of triangle ABC in a centre S outside the triangle

Step 1: I map vertex A (a line through S, transfer the distance) → A′.

Step 2: the same B → B′ and C → C′.

Step 3: I join A′B′C′.

A congruence check: |A′B′| must be the same as |AB| — remeasure with a compass. And side A′B′ is parallel to AB, it just points the opposite way.

The resulting triangle is the same size, turned upside down. Three lines, three transferred distances — the construction needs no more.

Which traffic signs and letters are point-symmetric?

A test by turning 180° in your head (or on tracing paper):

Letters: H, I, N, O, S, X, Z yes — turned they look the same. A, E, T no — they end upside down and different.

Digits: 0 and 8 yes, 6 and 9 swap with each other when turned (that is why they are underlined on dice).

A round sign with a symmetric pictogram yes; a triangular warning sign no — a triangle is never point-symmetric.

This test trains the eye: you look for a point around which the shape can be spun into itself.

Play it

Before you go to the sheet, try it with your eyes. This is not a test — it is a game. Tap until you see it.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Mulai with ten. When it goes well, add more.

Line symmetry of letters. Draw point symmetry on paper.

Practise the questions

On paper: turning half a turn

Draw with a compass and a ruler, transfer distances with a compass, not by guessing.

1. Choose a point S and a point A at a distance of 4 cm. Construct the image A′ and check that S bisects segment AA′.

2. Draw a triangle and map it in a centre S sitting outside the triangle. Remeasure two pairs of matching sides.

3. Write a large alphabet and circle the point-symmetric letters. Separately list those that are line-symmetric but not point-symmetric.

4. Design your own simple logo that is point-symmetric (it works upside down). Check by turning the paper.

Now you 💪

1. I can construct the image of a point: a line through the centre, the same distance past the centre.
2. I map a whole shape point by point and I know the image is congruent with the original.
3. I tell point symmetry from line symmetry and I give an example of a shape for each.

What to take from this lesson

• Point symmetry = a turn of 180° around centre S.
• The image of a point: through the centre and equally far on the other side.
• The image is congruent with the original; segments map to parallel segments.
• A shape is point-symmetric when a 180° turn covers it with itself.
• Line symmetry reflects in a line, point symmetry turns around a point — two different things.

© 2026 Ing. Martin Polak / AlgoRhino · Ketentuan penggunaan konten