Maths

Congruent and similar shapes

We are going to look at:

Similar shapes

Similar figures are identical in shape, but not in size.

For example, two circles are always similar.

Two squares are always similar:

And two rectangles could be similar:

But will probably not be.

Similar figures

Look at these similar figures:

B is an enlargement of A. The lengths have doubled, but the angles have stayed the same.

Remember: For any pair of similar figures corresponding sides are in the same ratio and corresponding angles are equal.

Look at the diagram below. Figures abcd and ABCD are similar.

and

Therefore,   ie, the sides are in the same ratio.

We can also say that and

and therefore

These facts can be used when solving problems.

Question

The rectangles pqrs and PQRS are similar. What is the length of PS?

The answer is PS = 15.75cm

Here's how to work it out. We know that the sides are in the same ratio, so:

Include the numbers you already know from the diagram like this:

PS = (7 × 9) ÷ 4 = 63 ÷ 4 = 15.75 cm

Remember : Try to use the formula which has the 'unknown' at the top of the fraction.

Question

wxyz and WXYZ are similar figures. What is the length of XY?

The answer is 4.5cm. Here's how to work it out. Because the shapes are similar we can write

xy = (4 × 9) ÷ 8 = 4.5

Question

What is the size of angle ?

Remember that the angles in similar figures stay the same. So is 57°.

Similar triangles

We know that two shapes are similar if their corresponding angles are equal, and their corresponding sides in the same proportion. The same is true for similar triangles.

To prove that two triangles are similar, we have to show that one (not all) of the following statements is true:

1. The three sides are in the same proportion.

2. Two sides are in the same proportion, and their included angle is equal.

3. The three angles of the first triangle are equal to the three angles of the second triangle.

We have only shown two angles in the above triangles. If we calculate the missing angles, we find that angle is 55° and angle is 40°. Therefore, all corresponding angles are equal.

Question

Are triangles PQR and SQT similar? If so, which of the above statements applies?

This question is much easier if we draw the triangles separately.

QR corresponds to QS (QR = 2QS) and QP corresponds to QT (QP = 2QT). Angle Q is the same in both triangles. Therefore, they are similar, because two sides are in the same proportion and their included angle is equal (statement 2).

Notice that they are similar, even though they are 'mirror' images (of different sizes).

Now try this one.

Question

Are triangles ABC and DBE similar? If so, which of the above statements (1, 2 or 3) applies?

ABC and DBE are similar, because all of the corresponding angles are equal (statement 3).

If you found this difficult, remember that angle B is common to both triangles. The rules of parallel lines state that

(corresponding angles)

and that (corresponding angles).

Congruent shapes

If two shapes are congruent, they are identical in both shape and size.

Remember: Shapes can be congruent even if one of them has been rotated or reflected.

Question

Which of the shapes in the illustration above are congruent?

Did you get the following pairs?

• A and G
• D and I
• E and J
• C and H

Remember: Shapes can be congruent even if one of them has been rotated (as in A and G) or reflected (as in C and H).

The symbol means 'is congruent to'.

Two triangles are congruent if one of the following conditions applies:

1. Three sides are the same

The three sides of the first triangle are equal to the three sides of the second triangle (the SSS rule: Side Side Side).

2. Two sides and one angle are the same

Two sides of the first triangle are equal to two sides of the second triangle, and the included angle is equal (the SAS rule: Side Angle Side).

3. Two angles and one side are the same

Two angles in the first triangle are equal to two angles in the second triangle, and one (similarly located) side is equal (the AAS rule: Angle Angle Side).

4. Two sides in right-angled triangle are the same

In a right-angled triangle, the hypotenuse and one other side in the first triangle are equal to the hypotenuse and corresponding side in the second triangle (the RHS rule: Right-angled, Hypotenuse, Side).

Question

For each of the following pairs of triangles, state whether they are congruent. If they are, give a reason for your answer (SSS, SAS, AAS or RHS).

Pair 1

Pair 2

Pair 3

1. Yes. RHS
2. Yes. SSS
3. No. The side of length 7cm is not in the same position on both triangles. Therefore, it is not AAS.

Similar areas and volumes

We already know that if two shapes are similar their corresponding sides are in the same ratio, and their corresponding angles are equal.

Look at the two cubes below:

The cubes are similar, and the ratio of their lengths is a:b or

Question

What is the ratio of:

1. the area of their faces?
2. their volumes?

1. Cube 'a' has a face area of a2

Cube 'b' has a face area of b2

The ratio of their areas is a2 :b2 or

2. Cube 'a' has a volume of a3

Cube 'b' has a volume of b3

The ratio of their volumes is a3 :b3 or

For any pair of similar shapes, the following is true:

Ratio of lengths = a:b or

Ratio of areas = a2:b2 or

Ratio of volumes = a3:b3 or

Question

These two shapes are similar. What is the length of x?

The ratio of the areas is 25:36 (a2:b2)

The ratio of the lengths is a:b Therefore, we find the square roots of 25 and 36. Ratio of lengths = 5:6

5:6 = 2:x

So

(multiply both sides by 2) = x

x = 2.4cm

Now you try one.

Question

Two similar pyramids have volumes of 64cm3 and 343cm3. What is the ratio of their surface areas?

The answer is 16:49. Here is how to work it out. Try to fill in the blanks below:

Ratio of volumes = 64:?

To find the ratio of the lengths, we find the cube roots of 64 and ?

Therefore, the ratio of the lengths is ?:7

To find the ratio of the areas, we square ? and 7

The ratio of the areas is 16:49.

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