Turn it, flip it

The idea

While checking for congruence, a figure may be rotated or flipped before superimposing it on the other figure.

Is a turned figure still a twin?

Back to the signboard. The symbol on the second board is drawn turned round, or even flipped over like a mirror image.

Will the tracing-paper copy of the first symbol still be able to fit it?

• Yes, if we may turn and flip the paper
• No, a turned figure is a different figure
• Only if we only slide it

Slide, turn, flip

The paper copy can now be dragged, turned (⟲ ⟳, 30° each) and flipped over (⇋). Find out which pairs are twins.

Moving a figure does not change it

Three pairs fitted after a turn or a flip, and each time it was the same figure. The last pair never fitted, however we moved the paper.

While checking for congruence, a figure may be rotated or flipped before it is superimposed on the other figure.

• Move | Does the figure change?
• Slide | No
• Turn | No
• Flip over | No

Notes

While checking for congruence, a figure may be rotated or flipped before it is superimposed on the other figure.

Check yourself

A left glove is turned over and laid on a right glove. They fit exactly. Are the two gloves congruent?

The second arrow is the first arrow turned half way round. Are the two arrows congruent?

Triangle PQR has sides of 3 cm, 4 cm and 5 cm. It is flipped over to make triangle P′Q′R′. How long is the longest side of P′Q′R′, in cm?

Answer: 5

Flipping only moves the triangle. It does not change any length, so the sides are still 3 cm, 4 cm and 5 cm. The longest is 5 cm.

Which of these changes means a figure is no longer congruent to the original?

• Yes. A flipped figure can fit exactly, so they are congruent. — correct. Yes! When we check congruence, a figure may be flipped over before we lay it on the other one.
• No. A left glove and a right glove face different ways.. Facing a different way is just a flip. Congruence allows a flip before superimposing.
• No. We are not allowed to turn the glove over.. We are allowed. Turning and flipping are allowed when we check for congruence.
• No. They point in different directions.. Pointing a different way is only a turn. A turned figure is still congruent to the original.
• No. One is on the left and the other on the right.. Where a figure sits makes no difference. We can slide and turn it to fit.
• Yes. Turn one half way round and it fits over the other. — correct. Yes! A turn does not change the shape or size.
• Turning it a quarter way round. A turn only changes how it faces. The figure is still congruent.
• Flipping it over like a page. A flip does not change the shape or size. The figure is still congruent.
• Stretching it to double the length — correct. Yes! Stretching changes the size and shape. Sliding, turning and flipping do not.