Constructing Triangles

8098 /www/nrich/html/content/id/8098/ 1 0000-00-00T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">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: <video controls="controls" height="300" src="Constructing%2520triangles.mp4" tabindex="0" width="400"> </video> 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: <ul> <li>Can you draw more than one triangle from each set of three numbers?</li> <li>When is it possible to construct a triangle from the three numbers generated?</li> <li>Is there a quick way to tell if it will be possible to construct a triangle?</li> </ul> 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. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 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! But the key questions still remain: 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? Do send us your thoughts.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> This problem not only gives an engaging context in which to practise drawing triangles with ruler and compasses, but also prompts students to think about the geometry of 'impossible' triangles, congruence, and perhaps even permutations, combinations and probability. <h3>Possible approach</h3> 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: <ul> <li>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? </li> <li>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? </li> <li>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?</li> </ul> Students could then move on to analyse what happens when they roll a die 4 times to generate numbers to draw quadrilaterals. <h3>Possible extension</h3> 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? <h3>Possible support</h3> <a href="/5045">Sticks and Triangles</a> introduces 'possible' and 'impossible' triangles with a more straightforward approach using matchsticks. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Take a look at the problem <a href="/5045">Sticks and Triangles</a> for an introduction to the ideas in this problem. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><a href="/content/id/8098/constructing%20triangles.xls">constructing triangles.xls</a> 34 out of the 56 combinations make triangles 111 out of the 216 possible rolls make triangles</mdoxml> 2 3 0 0 1 0 0 Constructing Triangles Generate three random numbers to determine the side lengths of a triangle. 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