Constructing triangles
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
Problem
Constructing Triangles printable sheet
Take a ten-sided die (or other random number generating tools - a pack of cards with the picture cards removed, a calculator, a phone app...) and generate three numbers. Construct a triangle using these three numbers as the side lengths.
If you're not sure how to use a ruler and compasses to construct a triangle given the lengths of its three sides, watch the video below:
Generate a few more sets of numbers and draw some more triangles.
What do you notice?
Here are some questions you might like to consider:
- Can you draw more than one triangle from each set of three numbers?
- When is it possible to construct a triangle from the three numbers generated?
- Is there a quick way to tell if it will be possible to construct a triangle?
Here is a game you could play:
Start with 10 points. Roll three dice. If a triangle can be drawn, you gain a point, if it can't, you lose a point. If you reach 20 points you win the game, if you reach 0 you lose.
Which is the more likely result?
Here is a game you could play with another person:
Player A chooses an integer length between 1 and 10cm. Player B randomly generates the lengths of the other two sides. If a triangle can be drawn, Player B wins; otherwise they lose.
Is there a "best" length that Player A should choose?
Is this a fair game?
Now explore what happens if you generate 4 numbers and draw a quadrilateral.
Getting Started
Take a look at the problem Sticks and Triangles for an introduction to the ideas in this problem.
Student Solutions
Eugena from City of London School for Girls made a good start to this problem:
Say you roll three dice and you end up with the numbers 6, 3 and 1.
You draw the 6cm line and then with a compass and ruler you measure 3cm.
After you draw the arc, you repeat the same step, the only change being that
instead of measuring 3cm you measure 1cm.
Draw the arc and you'll see that the two arcs will not intercept.
The three numbers didn't work...
Say you roll three dice and you end up with the numbers 5, 3 and 6.
You draw the 5cm line and then with a compass and ruler you measure 3cm.
After you draw the arc, you reapeat the same step, the only change being that
instead of measuring 3cm you measure 6cm.
Draw the arc and you'll see that this time the arcs intercept.
The three numbers worked!
The 5/6 Maths Extension (MEP) group from Lumen Christi School in Australia also trialled some side lengths and made the following observation:
We discovered sets of lengths for which we were able and unable to make triangles. The ones that worked were the ones where the two shorter lengths added up to more than the longest length. For the ones that didn't work we found that the two smaller lengths didn't add up to the length of the longest side.
What we weren't sure about was what if the lengths of the two shorter sides add up to the longest length. We tested that and found out that they didn't make a triangle. Why it is so: If the two smaller lengths of the triangle add up to the length of the longest side, the two sides intersected only when they met at the line. So it didn't make a triangle.
Well done for spotting this special case and for considering it very carefully!
Sam, Yordan, Isla from Kings Ely, Elijah from Reading School and Amy from Melbourn Village Collage formulated a quick way to tell if it will be possible to construct a triangle from the three numbers generated. Yordan says:
Triangles can only be constructed if the sum of both smallest sides is greater than the longest side.
If $s$ and $c$ were the shorter sides, and $l$ was the longest, you could only construct a triangle if:
$$ l < s + c\;.$$
Members of the Senior Maths Challenge Group at Lyneham Primary School in Australia came to the same conclusion. They added:
If you roll a '1', the others will have to be the same.
Amy also remarked:
It's not possible to draw more than triangle from each set of measurements; even if it looks like you have you have just flipped or rotated it.
Well done to everyone!
Susan claims that she used the spreadsheet here to decide whether you are more likely to win or to lose in the one player game. Do you see how?
Do send us your thoughts.
Teachers' Resources
Why do this problem?
This problem offers an engaging context in which to practise drawing triangles with ruler and compasses, prompts students to think about the geometry of 'impossible' triangles and congruence, and then challenges them to think about permutations, combinations and probability.
Possible approach
Students will need to be aware of how to construct a triangle given three sides.
Students will need plain paper, ruler and compasses. They will also need dice (ideally 10-sided) or other random number generating tools (a pack of cards with the picture cards removed, a calculator, a phone app...).
"In your pairs, choose one person to go first. Generate three random numbers. Your partner has to construct triangles whose three side lengths are given by the three numbers. They get one point for each different triangle they manage to draw. Then swap over. Whoever has the most points after 5 goes each is the winner. Keep a record of the numbers you generate."
(By introducing the task in this way, you might prompt students to consider whether there are different triangles with the same three side lengths, rather than moving straight on when they have constructed one.)
Give students some time to play the game, then bring the class together and invite them to share any thoughts they had.
Possible prompts if thoughts are not forthcoming:
"Does anyone have examples where they could draw more than one different triangle?" - this could draw out a discussion on congruence and what it means for two triangles to be 'different'.
"Does anyone have examples where they couldn't draw a triangle at all?"
Divide the board in half and collect together examples where triangles can and can't be drawn.
"With your partner, see if you can come up with a convincing explanation why it is possible to draw triangles with these side lengths, but impossible to draw triangles with these side lengths."
"In a few moments, I'm going to choose three large numbers and you will need to be able to explain straight away whether we could draw a triangle using those three numbers as the side lengths, or not."
Once students have had a chance to discuss, bring the class together and choose sets of three numbers such as {35, 43, 79} or {12, 23, 32} and invite a selection of students to explain clearly whether a triangle could be drawn with each set. Finally, come up with a clear statement as a class to explain how to determine whether a triangle can or can't be drawn.
Playing and analysing these three games would be suitable follow-up activities:
- Player A chooses an integer length between 1 and 10cm. Player B randomly generates the lengths of the other two sides. If a triangle can be drawn, Player B wins; otherwise they lose. Take it in turns to be Player A. Is there a "best" length that Player A should choose?
- A game for two: Player A randomly generates the "first" side. Player B randomly generates the other two sides. If a triangle can be drawn, Player B wins a point; otherwise Player A wins a point. First to reach 20 points wins the game. Is this a fair game?
- A solo game: start with 10 points. Roll three dice. If a triangle can be drawn, you gain a point, if it can't, you lose a point. If you reach 20 points you win the game, if you reach 0 you lose. Which is the more likely result?
Possible support
Sticks and Triangles introduces 'possible' and 'impossible' triangles with a more straightforward approach using matchsticks.
Possible extension
Students could be asked to consider the angle properties of the triangles that can be made:
Is it possible to predict whether a triangle will be right angled, acute angled or obtuse angled, simply by knowing the three sides?
Students could analyse what happens when they roll a die 4 times to generate numbers to draw quadrilaterals. They could be encouraged to explore the range of possibilities using dynamic geometry software such as GeoGebra.