Isosceles Triangles

2666 /www/nrich/html/content/id/2666/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">This question is about isosceles triangles with an area of 9 cm$^2$. Each vertex of the triangle must be at a grid point of a square grid <div style="float: left;"> </div> (all the vertices will have whole number coordinates).<mdo:image alt="" src="Isosceles.gif" style="width: 108px; height: 235px; float: right;"></mdo:image> One of the vertices must be at the point (20, 20). The picture shows one example. How many different triangles satisfy these conditions? <div> Try to draw them all. You may wish to use the interactivity below.</div> Can you explain how you know that you have found them all? <a href="http://nrich.maths.org/content/id/2666/isosceles2.swf">Full Screen Version</a> <mdo:flash height="450" id="/content/id/2666/isosceles2.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param> <param name="movie" value="/content/id/2666/isosceles2.swf"></param> <param name="flashplayerversion" value="6"></param> <param name="height" value="450"></param> <param name="width" value="550"></param> </mdo:flash> <a href="http://nrich.maths.org/public/viewer.php?obj_id=5588&part=">Click here for a poster of this problem</a>. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Yanqing from Devonport High School for Girls sent us the only correct solution to this problem. She wrote: First, I worked out how many types of isosceles triangles there are that have an area of 9cm², and can be drawn with integer co-ordinates. I found 3 types: base = 18, height = 1, base = 6, height = 3, base = 2, height = 9. (These would not work the other way round because it would be impossible to have integer co-ordinates.) I then worked out that because there are 3 vertices on a triangle, and there are 4 directions in which a triangle could go, it would be possible to have 12 ways of placing one type of triangle around the point (20,20): you can place it 3 ways upwards, 3 ways to the left, to the right and down. I then tried this out and found it to be the case. Because there are 3 types of isosceles triangles, and 12 ways to place each one, it is possible to draw 36 isosceles triangles with integer co-ordinates and one vertex on (20,20) which have an area of 9cm². Well done Yanqing - this is a very clear explanation. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <div style="font-style: italic;">Paul Andrews, a respected mathematics educator based at Cambridge University, explains why he likes <a href="http://nrich.maths.org/public/viewer.php?obj_id=2666">this problem</a> :</div> <div style="font-style: italic;"></div> <blockquote>"I first saw it used as a homework question more than ten years ago with a year seven class (11 year old students) in Budapest and have used it with every group I have taught since - initial teacher training, mathematics education research students, in-service; everyone gets it. I love it because it forces an acknowledgement of so many different topics in mathematics and is sufficiently challenging to keep almost any group meaningfully occupied within a framework of Key Stage Three mathematics content. This, for me, is the key to a good problem; Key Stage Three content alongside non-standard and unexpected outcomes. For example, notwithstanding the obvious problem solving skills necessary for managing such a non-standard problem, I think it requires an understanding of coordinates, isosceles triangles and the area of a triangle. It requires an awareness of the different factors of 18 and which are likely to yield productive solutions. It requires, also, an understanding not only of basic transformations like reflection and rotation but also an awareness of their symmetries. Moreover, the solution, which is numerically quite small, is attainable without being trivial. In short, I love this problem because of the wealth of basic ideas it encapsulates and the sheer joy it brings to problem solvers, of whatever age, when they see why the answer has to be as it is. It is truly the best problem ever and can provoke some interesting extensions."</blockquote> <h3>Possible approach</h3> <div>The richness of this task might best be exposed by working on the problem in small groups. The groups should be encouraged to keep asking the questions: are our solutions valid? have we found all of the solutions?</div> <div>Once groups feel that they have finished, they could try to explain clearly to the class why they have found all of the solutions. Those who missed out solutions could be encouraged to think about why these solutions were overlooked.</div> <h3>Key questions</h3> <ul> <li>How do you know that your triangles have the correct area?</li> <li>Are there any more like that?</li> <li>Can you explain why you are certain that there are no more solutions?</li> </ul> <h3>Possible extension</h3> <div>Repeat for isosceles triangles of areas $50$ and $15$. Which starting areas lead to more/fewer solutions? Is the number of solutions predictable?</div> <div>and/or</div> <div>The problem doesn't insist that one edge is horizontal/vertical - what new triangles can be found, with all sides sloping? Can you prove that you have got them all?</div> <h3>Possible support</h3> <div>The problem can be used as a context for practising: drawing isosceles triangles, using co-ordinates accurately, calculating areas, communicating results, working with others, etc.</div> <div>When the students have worked out the basic possibilities for the isosceles triangles, they could cut them out to help to search for congruent solutions.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Isosceles triangles have two equal sides. The area of a triangle is half the base times the height. Any of the three vertices of the triangle can be at $(20, 20)$. There are sets of identical (congruent) triangles with an area of $9$cm$^2$. How many triangles are there in each set? How many sets? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Yanqing from Devonport High School for Girls sent us the only correct solution to this problem. She wrote: First, I worked out how many types of isosceles triangles there are that have an area of 9cm², and can be drawn with integer co-ordinates. I found 3 types: base = 18, height = 1, base = 6, height = 3, base = 2, height = 9. (These would not work the other way round because it would be impossible to have integer co-ordinates.) I then worked out that because there are 3 vertices on a triangle, and there are 4 directions in which a triangle could go, it would be possible to have 12 ways of placing one type of triangle around the point (20,20): you can place it 3 ways upwards, 3 ways to the left, to the right and down. I then tried this out and found it to be the case. Because there are 3 types of isosceles triangles, and 12 ways to place each one, it is possible to draw 36 isosceles triangles with integer co-ordinates and one vertex on (20,20) which have an area of 9cm². Well done Yanqing - this is a very clear explanation. </mdoxml> 2 3 0 0 1 0 0 Isosceles Triangles Draw some isosceles triangles with an area of 9 cm squared and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw? Using, Applying and Reasoning about Mathematics Visualising Measures and Mensuration Area 2D Geometry, Shape and Space Isosceles triangles Coordinates and Coordinate Geometry Coordinates - first quadrant Using, Applying and Reasoning about Mathematics Working systematically Information and Communications Technology Interactivities Secondary Mapping Document Area and volume LS Secondary Mapping Document DisplayCabinet Secondary processes PM - Working Systematically