Mathematics Year 7: Similar figures, ratio of similarity and homothety
Mathematics, Year 7 (Portugal): Recognise figures with the same shape, find the ratio of similarity, build enlargements and reductions by homothety and read map scales.
Same shape, different size
In everyday talk, ‘similar’ means ‘alike’. In maths the rule is exact: two figures are similar when one is an enlargement or a reduction of the other, so they have the same shape. Matching angles are equal, and matching lengths are all multiplied by the same number k, the ratio of similarity (a constant of proportionality). A 4 cm by 6 cm rectangle and a 6 cm by 9 cm rectangle are similar: 6 ÷ 4 = 9 ÷ 6 = 1.5. Going the other way, the ratio is 4 ÷ 6 = 2/3, a reduction.
How to build an enlargement: the homothety
Choose a point O, the centre, and a number k greater than 0. For each vertex P, mark P' on the ray OP so that OP' = k × OP. Then join the points P' in the same order. With k = 2.5 and OA = 3 cm, you get OA' = 7.5 cm. If k > 1 it is an enlargement, if k < 1 a reduction. The image keeps the angles and every side is multiplied by k, so it is similar to the original, with ratio of similarity k.
Two maps of the same route
A route measures 6 cm on a map at scale 1:25 000. The scale says 1 cm on the map is 25 000 cm on the ground. Real distance: 6 × 25 000 = 150 000 cm = 1500 m = 1.5 km. On another map of the same area at scale 1:100 000, the route measures 150 000 ÷ 100 000 = 1.5 cm. The two maps are similar figures: the second is a reduction of the first with ratio 1.5 ÷ 6 = 1/4, which is 25 000 ÷ 100 000. A scale is the ratio of similarity between the map and the ground.
The trap: adding instead of multiplying
To enlarge a 4 cm by 6 cm rectangle, it feels natural to add 2 cm to each side: 6 cm by 8 cm. But 6 ÷ 4 = 1.5 while 8 ÷ 6 ≈ 1.33: the sides were not multiplied by the same number, so the new rectangle is stretched and not similar to the first. The same happens when you widen a photo without changing its height. Also, equal angles are not enough for a rectangle: all rectangles have four right angles, and they are not all similar.
Where you see this outside school
An A3 sheet is an enlargement of an A4 sheet: 21 × 29.7 cm becomes 29.7 × 42 cm, and 29.7 ÷ 21 ≈ 42 ÷ 29.7 ≈ 1.41, which is why photocopiers offer 141%. Plans, models, photos and maps are similar to what they show. When you zoom a map on your phone you change the scale, not the shape. And an image stretched in one direction only stops being similar: it looks squashed.
Test what you learned
Which statement describes two similar figures?
- a) They have the same area
- b) They have the same perimeter
- c) They have the same number of sides
- d) One is an enlargement or reduction of the other: equal angles, and matching lengths all multiplied by the same number
Correct answer: d) One is an enlargement or reduction of the other: equal angles, and matching lengths all multiplied by the same number — Right: same shape, any size. The number that multiplies the lengths is the ratio of similarity.
A path measures 10 cm on a map at scale 1:20 000. How long does it measure on a map of the same area at scale 1:50 000?
- a) 25 cm
- b) 4 cm
- c) 2 km
Correct answer: b) 4 cm — Right: the real length is 10 × 20 000 = 200 000 cm, and 200 000 ÷ 50 000 = 4 cm. The ratio between the maps is 20 000 ÷ 50 000 = 2/5.
A 4 cm by 6 cm rectangle is ‘enlarged’ by adding 2 cm to each side, giving 6 cm by 8 cm. Are the two rectangles similar?
- a) Yes, the same 2 cm was added to every side
- b) Yes, both have four right angles
- c) No: 6 ÷ 4 = 1.5 but 8 ÷ 6 ≈ 1.33, so the sides were not multiplied by the same number
- d) No, because their areas are different
Correct answer: c) No: 6 ÷ 4 = 1.5 but 8 ÷ 6 ≈ 1.33, so the sides were not multiplied by the same number — Right: the new rectangle is stretched. A true enlargement would keep the ratio, for example 6 by 9, with 1.5 on both sides.
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