Equal Equilateral Triangles
Imagine you have lots of copies of the two triangles pictured below:
You may wish to print off and cut out these Yellow and Green triangles.
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Can you make a regular hexagon using only yellow triangles ?
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Can you make a regular hexagon using only green triangles ?
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Can you make a regular hexagon from yellow triangles that is the same size as a regular hexagon made from green triangles ?
Did that tell you something about yellow and green triangles, about how they relate to each other ?
- Can you make an equilateral triangle from yellow triangles ?
- Can you make an equilateral triangle from green triangles ?
- Can you make an equilateral triangle from yellow triangles that is the same size as an equilateral triangle made from green triangles ?
When you've got inside this problem and started to feel your way around, please do take a look at Impossible Triangles? - it's the perfect follow on. There's even a video to watch to make getting started easy.
Printable NRICH Roadshow resource.
Here are some hexagons you could use as a starting point to explore other possibilities:
The screen shot on the Hints page shows some solution arrangements.
When we looked at the solutions sent in, some people had started by putting triangles together until they could make a hexagon, others started with a hexagon and looked for ways it might split up into the two types of triangle.
Well done to Daniel from Royston and to Georgina from St. George's School for seeing the answer to the first important question :
'Did that tell you something about yellow and green triangles, about how they relate to each other' ?
Two yellow triangles make a rhombus of a size which can also be made by two green triangles.
And that lets us see that the yellow and green hexagons are equal because the yellow and green triangles have the same area (half that rhombus) and both hexagons use six triangles. But the equilateral triangles (below) do not use the same number of triangles and so cannot be the same size.
Esther had a great approach using sequences :
You can make equilateral triangle arrangements from yellow triangles :
The numbers needed follow the sequence : 1, 4, 9, 16 . . . a ² where a is the number in the sequence.
When I tried making equilateral triangles with the green triangles I could see that one way to do it was by replacing each equilateral space with three green triangles ( which would also of course scale up the size of my arrangement ).
When working with green triangles the numbers follow the sequence: 3, 12, 27, 48 . . . 3b ² where b is the number in the sequence.
Which means I'm looking for a value of a and of b so that a ² = 3b ²
Keeping in mind that a and b are whole numbers ( a position in a sequence ) , a ² would have to have a factor of 3 to match 3b ² , but any factor a ² has it will have twice because it's a number squared. So a ² can only contain an even numbers of 3 s in its prime factors and 3b ² can only contain an odd numbers of 3 s. This means that a ² and 3b ² are never going to match, and an equilateral arrangements made from yellow triangles is never going to match an equilateral arrangements made from green triangles
This is great reasoning, but how do we know that replacing each equilateral space with three green triangles is the only way to build up to an equilateral triangle ?We hope you enjoy the brain-stretching this kind of reasoning involves.Be encouraged, students a lot older than Stage 4 still wrestle with this sort of thinking.
Why do this problem?
The bisected equilateral triangle (30-60-90) is an important shape for students to become familiar with, and the early questions in this problem draw out the compound forms built from this triangle.
The final question, to which the other questions serve as an introduction, explores surd forms and the circumstances when one length can, or cannot, be a rational multiple of another.
Possible approach
Key questions
- While looking at the screen shot on the Hints page : are the green and yellow hexagons really the same size or just close ? How can we be sure of that? [this draws out the relationship between the yellow and green triangles, including that they have equal area - draw attention to the line of symmetry for each shape to help students visualise this relationship]
- Still looking at the screen shot on the Hints page : can the larger hexagon be made from green triangles ? If so make it, or if not explain why this cannot be done.
Possible extension
The second of the key questions above is challenging, able students should be encouraged to reason this thoroughly.
The following problems make a natural follow on from this activity: The Square Hole and Impossible Triangles?
Possible support
There are a number of activities which can provide valuable auxiliary experiences for students working on this problem :
- Drawing first the equilateral triangle using only a straight edge and compasses, and then creating the isosceles triangle, likewise, will give a strong sense for the symmetry of each triangle and the relationship between them.
- For students who are not able to make any progress with grasping the last question, 'Can you make an equilateral triangle from yellow triangles that is the same size as an equilateral triangle made from green triangles ?' the earlier questions, although a lead up to this, nevertheless make a very good activity in themselves. Some children 'play' a long time with cut out triangles arranged on a table, motivated by the aesthetic appeal of emerging pattern possibilities, this is excellent grounding for other mathematical ideas to be built on.