Number the sides
The triangles in these sets are similar - can you work out the
lengths of the sides which have question marks?
Age
7 to 11
Challenge level
Problem
The triangles in this set are 'similar':
Image
'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:
Image
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:
Image
Set 3:
Image
Set 4:
Image
Set 5:
Image
Set 6:
Image
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
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
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
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
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
Perhaps you can expand
on what Sophie has said? What we're interested in is why we would
"times" by the amounts that Sophie has suggested. Emma from Lord
Williams School talked about patterns which she could see in the
lengths of the sides. This is a good way to look at it, Emma.
Daniel from Reading
School sent in pictures of the triangles with their lengths
labelled:
Image
Image
Image
Image
Teachers' Resources
Number the Sides
The triangles in this set are 'similar':
Image
'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:
Image
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:
Image
Set 3:
Image
Set 4:
Image
Set 5:
Image
Set 6:
Image
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
You might suggest that children have a go at the Matching Triangles problem before they try this, which would offer a good basic introduction to similar triangles and might provoke some interesting discussion amongst the class.
This problem would be best introduced to the whole group at first. You could simply show them the first three triangles and ask them what they think the missing length is. Invite children to explain to everyone how they worked out their response. Listening to different ways of articulating the thought processes will help those who are not so sure find an explanation which they can make their
own. The next step might be to show the group the same set of triangles but with the third triangle in a different orientation as in the second image. This will challenge them a little at first but makes a good lead into the main activity.
You could print off copies of this sheet for the children to use, which has all the sets of triangles on it.
Key questions
Would it help to write out the lengths of the sides of each triangle in a set?
Why don't you compare the shortest side of the first triangle with the shortest side of the third triangle?
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?