Number the Sides
2 out of 3
The triangles in these sets are similar - can you work out the lengths of the sides which have question marks?
Problem
The triangles in this set are 'similar':
'Similar' means that the triangles are exactly the same shape, but not the same size. The sides are in the same ratio to each other. (Note that these triangles are not drawn to scale.)
What can you say about the length of the side of the third triangle which is marked with a question mark? Of course the triangles could be different ways up, too:
There are five more sets of similar triangles below. Can you work out the lengths of the sides marked with a question mark?
Set 2:
Set 3:
Set 4:
Set 5:
Set 6:
Getting Started
It might help to write out the lengths of the sides of each triangle in a set.
In the first set, try comparing the shortest side of the first triangle with the shortest side of the third triangle. What do you notice? How about comparing the two "middle length" sides?
Can you use this to work out what the longest side length is in the third triangle?
Student Solutions
Many of you sent in answers to this problem, but not many of you explained how you arrived at the answers. It can be hard to put into words what you did, but if you try to explain, you often find you end up understanding what you did much better.
Sophie from Haywards Primary School was one of the few people who gave us some detail about what she did. She says:
Set one:
Times the first length of the triangle by 3 so the answer is 9
Set two:
Times the first length of the triangle by 2 for the second and then add it again to get the answers of the third triangle so the answer is 8, 12 and 15
Set three:
All the lengths are the same so the answer is 2, 2, 4 and 4
Set four:
Two thirds of the first length for the second triangle and then one third for the third triangle so the answer is 6, 4, 3 and 3
Set five:
For the first triangle you half the length of the second triangle and you add the length that is half of the second triangle so the answer is 2, 4, 6 and 15
Set six:
Times the 3 to get to 9 so you times the 4 to get 12 so the answer is 9, 9 and 12
Times the 4 by 2 to get 8 so times the 3 by 2 you get 6 so the answer is 8, 6 and 6
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This problem is a good introduction to the numerical aspects of similar triangles. It will also bring in ratio, and use multiplication and division.
Possible approach
Key questions
Possible extension
Learners could draw their own sets of similar triangles.
Possible support
Suggest using the interactivity in this simpler problem which introduces similar triangles.