Constructing Triangles

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."

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.