Maths

Congruent and similar shapes

We are going to look at:

Similar shapes

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

image: circle

For example, two circles are always similar.

image: 2 squares

Two squares are always similar:

image: 2 rectangle

And two rectangles could be similar:

image: 2 rectangle

But will probably not be.

Similar figures

Look at these similar figures:

Shape A: Five sided shape, from clockwise directions mesurements from top: 2cm, left-angle: 225 degrees, right-side: 4cm, bottom-left corner: 45 degrees, bottom: 5cm, left-side: 7cm. Shape B: Five sided shape, from clockwise directions lengths from top: 4cm, left-angle: 225 degrees, right-side: 8cm, bottom-left corner: 45 degrees, bottom: 10cm, left-side: 14cm.

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.

images: 4 sided shape: left shape: top length: corners BC:4cm, right-side: CD: corners CD: both 90 degree angles, bottom length: DA: 8cm, left-side: at an angle: AB. Right shape: top length: corners BC:6cm, right-side: CD:  corners BC:4cm, right-side: CD, corners CD: both 90 degree angles, bottom length: DA: 12cm, left-side: at an angle: AB

ad over AD = 8 over 12 = 2 over 3 and bc over BC = 4 over 6 = 2 over 3

Therefore, ad over AD = bc over BC   ie, the sides are in the same ratio.

We can also say that AD over ad = 12 over 8 = 3 over 2 and BC over bc = 6 over 4 = 3 over 2

and therefore AD over ad = BC over bc

These facts can be used when solving problems.

Question
image: two rectangles, left rectangle: top-left corner: Q, top-right: R, bottom-right: S, bottom-left: P, length QP: 4cm, PS: 9cm. Right rectangle: same labels, QP: 7cm, PS: ?.

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

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Answer

The answer is PS = 15.75cm

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

PS pver ps = PQ over pq

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

PS over 9 = 7 over 4

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
image: two four-sided shape: top-right: X, right: Y, bottom-left: Z, left: W, length WX: 8cm, XY: 4cms, corner X: 57 degrees. Right shape: same lettered labels: length WX: 9cm.

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

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Answer

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

9 over 8 = xy over 4

xy = (4 × 9) ÷ 8 = 4.5

Question

What is the size of angle wxy ?

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Answer

Remember that the angles in similar figures stay the same. Sowxy 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.

image: two triangle: left triangle: left-side: 3cm, right-side: 5cm, bottom: 4cm. Right triangle: left-side: 4.5cm, right-side: 7.5cm, bottom: 6cm.

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

image: two triangles: left triangle: right-side: 5cm, bottom-right corner: 30 degree, bottom: 3cm. Right triangle: right-side: 10cm, bottom-right corner: 30 degree, bottom: 6cm

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

image: two triangles: left triangle: top Y corner: 85 degrees, right Z corner: 40 degrees, left corner: X. Right triangle: same labels: Y: 85 degrees, X: 55 degrees.

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

Question

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

image: triangle: top corners: Q, right: T, right corner: R, left corner: P, left: S. Lengths: QT: 3cm, TR: 5cm, PS: 2cm, SQ: 4cm.

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Answer

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

image: two triangle: left triangle: top: Q, right: T, left: S. Lengths: QT: 3cm, SQ: 4cms. Right triangle: same labels: QR: 8cm, PQ: 6cm.

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?

triangle, from clockwise directions lengths from right: labels BE:4cm, labels EC: 2cms, bottom: labels CA: 5cm. left side have two labels, AD: opposite to EC through to the top B,

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Answer

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

BDE = BAC (corresponding angles)

and that BED = BCA (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.

image: various shapes labelled with letters

Question

Which of the shapes in the illustration above are congruent?

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Answer

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 identical 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).

Two triangles, left-side and right-side of both triangles are equal in length. Bottom lengths is of both triangles are equal in length

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).

image: Two triangles, the right-side lengths of both triangles are equal to each other. The left-sides of both triangles are equal to each other. The top corner angle of both triangles equal to 80 degrees.

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).

image: Two triangles, left-side of both triangle are equal to each other. Top corners of both triangles both equal 105 degrees.

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).

image: two triangles, bottom corners of both triangles equal 90 degrees, right-side lengths of both triangle equal to each other. Left-side lengths of both triangles equal to each other.

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

image: two triangle, top corner of left triangle is 90 degress, right side of triangle: 30mm, bottom of triangle: 58mm. Right triangle: right side: 5.8cm, left side: 3cm, bottom left corner: 90 degress

Pair 2

image: two triangles, left triangle: right-side length: 4cm, bottom length: 5.5cm, left-side length: 3cm. Right triangle: top length: 5.5cm, right-side length: 3cm, left-side: 4cm

Pair 3

image: two triangles: left triangle: bottom right corner: 60 degrees, bottom length: 7cm, bottom left corner: 40 degrees. Right triangle: top left corner: 60 degrees, left length: 7cm, top right corner: 40 degrees.

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Answer
  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:

image: two cubes: left cube: (a) is equals in lengths for width, height and depth. Right cube:(b) is equal in length for width, height and depth.

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

Question

What is the ratio of:

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

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Answer
  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 a to the power of 2 over b to the power of 2

  2. Cube 'a' has a volume of a3

    Cube 'b' has a volume of b3

    The ratio of their volumes is a3 :b3 or a to the power of 3 over b to the power of 3

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

Ratio of lengths = a:b or a over b

Ratio of areas = a2:b2 or a to the power of 2 over b to the power of 2

Ratio of volumes = a3:b3 or a to the power of 3 over b to the power of 3

Question

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

image: two 6 sides shape (hexagon): left hexagon: area equal to 25cm squared, bottom right length:2cm. Right hexagon: area:36cm squared, bottom right length: (x)

Answer

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 6 over 5 = x over 2

(multiply both sides by 2) 12 over 5 = 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?

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Answer

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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