Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Write the ratio 8 : 12 in its simplest form.
Answer
2 : 3. - What do the three angles of any triangle add up to?
Answer
180 degrees. - Solve x ÷ 4 = 2.5.
Answer
x = 10.
Words used in this lesson
- similar
- the same shape, not necessarily the same size. For example: a photo and its enlargement.
- scale factor
- the number you multiply a length by to get the matching length. For example: 12 ÷ 2 = 6.
- corresponding
- in the same position in each shape; matching. For example: the two shortest sides.
- ratio
- a comparison of two numbers by dividing. For example: 12 : 6 = 2 : 1, a scale factor of 2.
- congruent
- exactly the same shape and size. For example: scale factor 1.
Measuring a tree with a shadow
At noon a 1.5 m post casts a 2 m shadow. At the same moment a nearby tree casts a 12 m shadow. Is the tree about 8 m, 9 m or 18 m tall? Make a guess and give a reason before you read on.
Two triangles are similar when they have the same shape: corresponding angles are equal and corresponding sides are in the same ratio. That ratio is the scale factor: new length ÷ old length.
For triangles, one of these is enough: two equal angles (AA), three sides in the same ratio (SSS), or two sides in the same ratio with the angle between them equal (SAS). Similar triangles are the same shape but not always the same size; if the scale factor is 1 they are also congruent.
Set a scale factor, then stretch one side. Say what you expect the ratios to be before you move.
All three ratios are 2 and the angles match, so B is similar to A with scale factor 2.
Show me the steps
- Scale factor 2, extra 0: B has sides 12, 10, 8. All three ratios are 2 and the angles match, so B is similar to A.
- Scale factor 1.5, extra 0: B has sides 9, 7.5, 6. All three ratios are 1.5. Similar again.
- Scale factor 2, extra 1: B has sides 13, 10, 8. The ratios are 2.17, 2 and 2, so B is not similar to A and its angles differ.
- Scale factor 0.5, extra 0: B has sides 3, 2.5, 2. All three ratios are 0.5: a scale factor below 1 is a reduction, and it is still similar.
Two slips to watch for. Similar means multiply, not add: adding 2 to every side of 4, 5, 6 gives 6, 7, 8, whose ratios are 1.5, 1.4 and 1.33, so it is a different shape. And the scale factor is new ÷ old: a bigger triangle has a scale factor above 1 and a smaller one below 1.
Worked examples
The tree: the post is 1.5 m with a 2 m shadow; the tree's shadow is 12 m. How tall is the tree?
- The sun's rays hit the post and the tree at the same angle, and both stand upright, so the two triangles have equal angles (AA): they are similar.
- Scale factor from shadow to shadow: 12 ÷ 2 = 6.
- Tree = 1.5 × 6 = 9 m.
Triangle P has sides 8, 12 and 14 cm. A similar triangle Q has a shortest side of 20 cm. Find its other two sides.
- Scale factor: 20 ÷ 8 = 2.5 (the shortest side matches the shortest side).
- 12 × 2.5 = 30 cm and 14 × 2.5 = 35 cm.
A line parallel to one side
When D is on AB and E is on AC and DE is parallel to BC, the small triangle ADE is a copy of ABC. Slide D along AB, and compare the three ratios.
D is half way from A to B: AD ÷ AB, AE ÷ AC and DE ÷ BC are all 0.5.
Show me the steps
- D a quarter of the way from A to B: AD = 2.75, AE = 2.25 and DE = 2.5. Every ratio is 0.25.
- D half way: AD = 5.5, AE = 4.5 and DE = 5. Every ratio is 0.5.
- D three quarters of the way: AD = 8.25, AE = 6.75 and DE = 7.5. Every ratio is 0.75. At every position the angles at D and B stay equal, so ADE is similar to ABC.
Your go. The first step is given. In triangle ABC, the point D is on AB and E is on AC, with DE parallel to BC (the same set-up as the second figure). AD = 4 cm, AB = 10 cm and DE = 6 cm. Find BC.
- DE is parallel to BC, so the angles at D and B are equal, and the angles at E and C are equal. Triangles ADE and ABC are similar (AA).
- Find the scale factor from ADE to ABC, then BC. Then check.
Check the answer
Scale factor = AB ÷ AD = 10 ÷ 4 = 2.5. BC = DE × 2.5 = 6 × 2.5 = 15 cm.
Check before you go on
1. Triangle X has angles 50°, 60° and 70°. Triangle Y has angles 70°, 50° and 60°. What can you say about them?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Two triangles are similar with scale factor 3. A side of the small one is 4 cm. How long is the matching side of the large one, in cm?
3. A triangle has sides 5, 12 and 13 cm. A similar triangle has a shortest side of 15 cm. How long is its longest side, in cm?
Level 2 · Building
4. A 2 m pole casts a 3 m shadow. At the same time a flagpole casts a 15 m shadow. How tall is the flagpole, in metres?
5. Triangle one has sides 6, 9 and 12. Triangle two is similar and has sides 8, x and 16. What is x?
6. In triangle ABC, AB = 7 cm and BC = 6 cm. Triangle DEF is similar, with DE matching AB and EF matching BC. If DE = 10 cm, find EF to 1 decimal place.
Level 3 · Stretch
7. A photograph is 10 cm by 15 cm. It is enlarged so that its longer side is 36 cm. How long is the shorter side, in cm?
8. In triangle ABC, D is on AB and E is on AC, with DE parallel to BC, as in the second figure. AD = 6 cm, DB = 3 cm and DE = 8 cm. Find BC in cm.
Something to think about
A triangle has angles of 40° and 65°. A second triangle has angles of 65° and 75°. Are they similar? If the side opposite the 65° angle is 6 cm in the first and 9 cm in the second, what is the scale factor?
One way to think about it
Yes: the third angles are 75° and 40°, so the angles match in pairs (AA), and the 65° angles are the matching ones. The scale factor is 9 ÷ 6 = 1.5.
Exit ticket
9. Which pair of triangles must be similar?