KS3 · Maths

Similarity and similar shapes

Zoom in on a photo and it just gets bigger. Stretch one edge and everyone looks wrong. Same photo, so what's the difference?

Maths · Similar shapes

Enlarge it, then try to break it
6.5 cm13.0 cm4.0 cm8.0 cm3.0 cm6.0 cm37°37°90°90°OABCB′C′A′

OA 6.5 cm. OA′ 13.0 cm. AB 4.0 cm. A′B′ 8.0 cm. BC 3.0 cm. B′C′ 6.0 cm. ∠A 37°. ∠A′ 37°. ∠B 90°. ∠B′ 90°. Relationship: When A′, B′ and C′ each sit exactly twice as far from O as A, B and C do, every image length is 2 × its object length and every angle is unchanged. Slide A′ off that spot and the picture stops being an enlargement.

OA6.5 cmOA′13.0 cmAB4.0 cmA′B′8.0 cmBC3.0 cmB′C′6.0 cm∠A37°∠A′37°∠B90°∠B′90°Slide A′ along its ray

When A′, B′ and C′ each sit exactly twice as far from O as A, B and C do, every image length is 2 × its object length and every angle is unchanged. Slide A′ off that spot and the picture stops being an enlargement.

Triangle ABC is the object. A′B′C′ is its image, enlarged from the centre O. Before you touch anything, read the chips: how does each image length compare with its object partner, and each image angle with its object partner? Then drag A′ along its ray and see which chips stop agreeing.

Maths · Similar shapes

What makes two shapes similar?

Three students are arguing about what makes two shapes similar. Each one sounds very sure.

Which idea is closest to what you think right now?
How sure are you?

Maths · Similar shapes

Similar, congruent, or neither?

Decide for each pair. Angles first (same, and in the same order?), then lengths (same, in proportion, or not?). Tick every set the pair belongs to, or none.

  • A Similar
  • B Congruent
  1. A triangle with sides 6, 9, 12 and a triangle with sides 2, 3, 4 (one is turned round)
  2. A triangle with sides 3, 4, 6 and a triangle with sides 6, 9, 12
  3. Two triangles that both have sides 6, 9, 12, one flipped over
  4. A 5 by 3 rectangle and a 10 by 6 rectangle
  5. A parallelogram with angles 100°, 80°, 100°, 80° and a trapezium with angles 100°, 100°, 80°, 80°
  6. Two circles, one bigger than the other
  7. A regular pentagon and a bigger regular pentagon
  8. Two regular hexagons with the same side length

Finding missing lengths

Problem

Two shapes are similar. A 12 cm side on the small shape matches an 18 cm side on the big one. (a) What does a 13 cm side on the small shape become? (b) What matches a 7.5 cm side on the big shape? (c) For a different pair of similar shapes, one has sides 4 and 8 and the other has sides x and 20. Find x.

Maths · Similar shapes

How tall is the traffic light?

A person 156 cm tall casts a shadow 117 cm long. At the same time of day, a traffic light casts a shadow 3 m long. Estimate how tall the traffic light is.

  1. We can't measure the traffic light, so we use the person as a guide. Their height and shadow give us a multiplier that takes a shadow to a height.
  2. missing step
Which line is step 2?

Maths · Similar shapes

Triangle inside a triangle

Triangle ABE sits inside triangle ACD. B is on AC, E is on AD, and BE is parallel to CD. AB = 5 cm, BC = 10 cm and AE = 4 cm. A student works out AD and ED. One line goes wrong. Which one?

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Same shape, different size

What you need to know

  • An enlargement changes size. The image can be bigger or smaller than the object, which is the figure you start with.
  • To enlarge from a centre, multiply each vertex's distance from it by the scale factor, along the ray through that vertex.
  • Image lengths are the scale factor times the matching object lengths, but the angles do not change.
  • Have a goYour classmate is very confident. "Enlarge a 30°, 60°, 90° triangle by scale factor 3 and you get 90°, 180°, 270°." What do you tell them?

    The angles stay 30°, 60° and 90°. Only the lengths are multiplied by 3.

    Enlargement multiplies lengths, not angles. Tripling the angles of a triangle would make them sum to more than 180°.

  • A property that stays the same after a transformation is called invariant, so angles are invariant under enlargement.
  • Scale factor is the multiplier between matching lengths. You find it by dividing, and below 1 the shape gets smaller.
  • Have a goA shape is enlarged with a scale factor of 1/2. Is the image bigger or smaller, and what happens to a 10 cm side?

    Smaller. The 10 cm side becomes 5 cm.

    A scale factor less than 1 makes the shape smaller, and each length is multiplied by it: 10 × 1/2 = 5.

  • Shapes are similar when only their size differs: lengths in the same proportions, and the orientation may differ.
  • The link is multiplicative, not additive. Triangle A (8, 11, 14) is not similar to triangle T (6, 9, 12).
  • Have a goTriangle T has sides 6, 9, 12. Triangle A has sides 8, 11, 14. Work out 8 ÷ 6 and 14 ÷ 12. What does that tell you?

    8 ÷ 6 = 4/3 and 14 ÷ 12 = 7/6. Different multipliers, so A is not similar to T.

    Similar shapes need the same multiplier for every pair of matching sides, and adding 2 to each side of T doesn't give that.

  • Similar shapes have the same angles in the same order. Equal angles always give similar triangles, but not always shapes with four or more vertices.
  • All circles are similar, and so are regular shapes with the same number of sides.
  • Congruent shapes have the same angles and lengths. They are similar with scale factor 1.

The big picture

Similar shapes differ only in size: every length is multiplied by the same scale factor, and the angles stay the same, in the same order. Congruent shapes are the special case where the scale factor is 1.

Key points

1Enlarging multiplies every length by the same scale factor. Angles do not change.
2Two shapes are similar when the angles match in the same order and the lengths are in the same proportions.
3Scale factor = a matching pair of lengths divided, image over object. Multiply to go small to large, divide to go large to small.
4Congruent shapes have the same angles and the same lengths. They are similar with scale factor 1.

Worked example

Problem

Rectangle R is 4 cm by 6 cm. Rectangle S is 10 cm by 15 cm. Are R and S similar? If they are, what is the scale factor from R to S?

⚠ Watch out

Trusting matching angles on their own. For triangles that is enough, but for a shape with four or more vertices the angles also have to be in the same order, and the lengths still have to be in proportion.

🧠

Memory hook

Same angles, same order, same multiplier. If one of those three fails, the shapes are not similar.

✓

Check yourself

Without looking back: in your own words, why does a photo stretched by different amounts in each direction look wrong, while a photo that is simply zoomed in does not? Use the word "multiplier".

Flashcards

(14)
Object and image: what is the difference?
The object is the starting figure before a transformation. The image is the figure that results after it.
How do you find an image vertex when enlarging from a centre?
Multiply the vertex's distance from the centre by the scale factor. The image vertex lies on the ray from the centre through the object vertex.
After an enlargement, what changes and what stays the same?
Each image length is the scale factor times its object length. The angles stay the same.
What does "invariant" mean?
A property that has not changed after a transformation.
How do you calculate a scale factor?
Divide a pair of matching lengths. The order depends on which shape is the object and which is the image. A scale factor below 1 makes the shape smaller.
When are two shapes similar?
When the only difference is their size: lengths in the same proportions, and orientation may differ. The image of an enlargement is similar to its object.
Is the link between similar shapes additive or multiplicative?
Multiplicative. Triangle T (6, 9, 12) and triangle A (8, 11, 14) are not similar, because A's lengths are not in the same proportions.
Do equal angles always make two shapes similar?
For triangles, yes: the order of the angles cannot be switched. With four or more vertices the angles can be in a different order, and then the shapes are not similar.
Which shapes are always similar?
All circles, and regular shapes with the same number of sides.
How do you find a missing length in similar shapes?
Work out the scale factor from a matching pair you know. Multiply to go from small to large and divide to go from large to small. Or use the multiplier inside one shape.
Does changing the units affect whether shapes are similar?
No. A 6 m by 12 m rectangle is similar to a 10 ft by 20 ft rectangle.
How are congruent and similar linked?
Congruent shapes have the same angles and lengths and fit exactly by rotation, reflection or translation. They are similar with scale factor 1. A pair can also be neither.
Congruent, similar or neither: what is the order of checks?
Angles first. If they differ, or are in a different order, it is neither. If they match, compare lengths: the same means congruent, in proportion means similar, otherwise neither.
Nested similar triangles: how do you find corresponding lengths?
Draw the two triangles separately and match each edge by the angles at its two ends.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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