1. Similar Triangles and other Polygons

2. Learning Objective Success Criteria
To understand the criteria that make two triangles (or two polygons) similar
Success Criteria
I can identify similar triangles and explain why they are similar
I can work successfully with ratios in solving geometric problems
I can solve problems involving polygon similarity

3. Are these two triangles similar?
Explain your answer

4. An object is similar to another object if they are the same shape
One object is larger than the other by a scale factor

5. Bigger versions can exist…

6. Two triangles are similar if
one of them is larger than the other by a scale factor.

7. Two triangles… 𝑋𝑌 𝐴𝐵 = 𝑌𝑍 𝐵𝐶
If similar will mean that their sides will be in proportion, that is, the ratio of the lengths of the same sides is the same. Largest divided by smallest provides the scale factor.
𝑋𝑌 𝐴𝐵 = 𝑌𝑍 𝐵𝐶

8. Finding the scale factor×𝟑 7m 6m ×𝟑
21m
18m 4m ×𝟑
12m

9. Are these two triangles similar?
12cm
3cm
5cm
20cm
3cm
12cm
Explain your answer

10. Problem: Explain your answer Which two triangles are similar? 6 12 510 6 11 18 18 15
Explain your answer

11. Two triangles…

12. Two triangles…

13. Two triangles…

14. Two triangles…

15. Two triangles…

16. Two triangles…

17. Two triangles…

18. Two triangles…

19. Two triangles…

20. Similar triangles are also equiangular
Equiangular, meaning they share the same angles.

21. Problem: Which two triangles are similar? 60° 70° 70° 50° 40° 70°
Hint: What do the angles in a triangle add up to?

22. Similar Triangles - examples
Here are some common examples of similar triangles. Note the parallel sides in the first two examples.
Remember:
Equiangular means equal angles.

23. Similar Triangles - calculation
Identifying similar triangles is a skill, as you are not normally told this. You may need to use geometric reasons to prove similarity first.
Identify the two equiangular triangles, if possible, draw them as two separate triangles
Identify which sides are in the same relative position
Apply appropriate ratios to help calculate unknown sides
Be careful:
Some figures may overlap – identify carefully the lengths required

24. Problem:

25. Problem:

26. Problem:

27. Problem: =12.8𝑐𝑚 𝑌𝑍 𝐵𝐶 = 𝑋𝑌 𝐴𝐵 ∴ 𝑥 8 = 32 20 ∴ 𝑥= 32×8 20 𝑂𝑢𝑟 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠
We are asked to calculate side length x.
All angles are equiangular, therefore we have similar triangles.
𝑌𝑍 𝐵𝐶 = 𝑋𝑌 𝐴𝐵
∴ 𝑥 8 = 32 20
∴ 𝑥= 32×8 20
𝑂𝑢𝑟 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠
=12.8𝑐𝑚

28. Problem: Calculate the height of the tree.
This is done using the shadow length, and a known height of another object.
ℎ 2 = 84 12
∴ ℎ= 84×2 12 =14 𝑚
𝑂𝑢𝑟 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠

29. Similarity – other polygons
The same principles can be applied to any polygons that are similar:
Corresponding angles are equal
Corresponding sides are in proportion
Following the same process as with triangles, you can through geometric reasoning solve for unknown sides.
Remember:
Corresponding means ‘in the same position’.

31. Practice
From homework book
Page 199 Ex F: Similarity