Geometric twins

The idea

Figures that are exact copies of each other, having the same shape and size, are congruent; congruent figures can be superimposed so that one fits exactly over the other.

Can we copy a big symbol?

A signboard has a symbol: a bent arm with sides AB = 4 cm and BC = 8 cm. Another board needs an exact copy of it.

A friend says the copy on the second board is exact. How can we be sure?

• Lay a tracing-paper copy of the first symbol on top and see if it fits exactly
• Check that both symbols are about the same size
• Check that both symbols have the same colour

Does the copy fit?

A tracing-paper copy of the first figure lies on top of it. Drag it onto the second figure. If it fits exactly, it turns green. Decide for three pairs.

Same shape and same size

Only the first pair fitted. The second had the same shape but was bigger. The third had arms of the same lengths, but a different angle between them.

Figures that are exact copies of each other, with the same shape and size, are congruent. Congruent figures can be superimposed: one fits exactly over the other.

• The two figures | Twins?
• same shape and same size | Yes, they fit exactly
• same shape, different size | No
• same arm lengths, different angle | No

Notes

Figures that are exact copies of each other, with the same shape and size, are congruent. Congruent figures can be superimposed: one fits exactly over the other.

Check yourself

Rani has a star stencil 10 cm wide. A second stencil is the same star, but 15 cm wide. Are the two stencils congruent?

Which pair of figures is congruent?

Two squares are congruent. The first has a side of 5 cm. What is the perimeter of the second square, in cm?

Answer: 20

Congruent figures have the same size, so the second side is also 5 cm. Perimeter = 4 × 5 = 20 cm.

Two figures have arms of exactly the same lengths, 4 cm and 8 cm. Must the two figures be congruent?

• Yes. They have the same shape.. The same shape is not enough. The sizes must match as well, or the tracing paper will not fit.
• Yes. They have the same number of points.. The same number of points does not make them the same size. A bigger star is not a twin of a smaller one.
• No. They have the same shape, but not the same size. — correct. Yes! Congruent figures need the same shape and the same size. A 15 cm star cannot fit over a 10 cm star.
• A square with a side of 5 cm and a square with a side of 6 cm. Both are squares, but the sizes are different. They do not fit over each other.
• A square with a side of 5 cm and a rectangle 5 cm by 6 cm. One side matches, but the shapes are different. They do not fit over each other.
• Two squares, each with a side of 5 cm — correct. Yes! The same shape and the same size, so one fits exactly over the other.
• Yes. Same lengths means same figure.. The lengths are the same, but the angle between the arms can still change. The third pair in the play showed this.
• No. The angle between the arms can be different. — correct. Yes! With a wider angle between the same arms the figure has a different shape, so the paper cannot fit.
• Yes, if they are the same colour.. Colour has nothing to do with the shape and size of a figure.