Triangles all Around

2850 /www/nrich/html/content/id/2850/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> You might like to have a look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=2852&part=index"> Nine-Pin Triangles</a> before trying this problem. How many different triangles can you draw on a circular pegboard which has four equally spaced pegs? What are the angles of each triangle? If you have a six-peg circular pegboard, how many different triangles are possible now? What are their angles? How many different triangles could you draw on an eight-peg board? Can you find the angles of each? You may like to use the interactivity to try out your ideas. <a href="/content/id/2850/circleAngles.swf">Full Size Version</a> <a style="left: 550px ! important; top: 0px ! important;" title="Block this object with Adblock Plus" class="abp-objtab-044561728425193925 visible ontop" href="https://nrich.maths.org/content/id/2850/circleAngles.swf"></a><mdo:flash height="400" width="550"><param value="/content/id/2850/circleAngles.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param><param value="400" name="height" ></param><param value="550" name="width" ></param></mdo:flash> For printable sets of circle templates for use with this activity, please see <a href="http://nrich.maths.org/public/viewer.php?obj_id=6676&part=index"> Printable Resources page</a>. Many thanks to Geoff Faux who introduced us to the merits of the 9 pin circular geo-board. The boards, moulded in crystal clear ABS that can be used on an OHP (185 cm in diameter), together with a teacher's guide, are available from Geoff at <a href="http://www.education-initiatives.co.uk/">Education Initiatives</a> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Sam sent us her work on the problem, including the angles of all the triangles. Thank you Sam! Can you see how she avoided counting any twice? She first counted triangles with two corners on neighbouring pegs, then those two apart, and so on. She called two triangles the same if they had the same angles and not just if they used the same pegs. She explains this in a bit more detail below. Here is her work: First I labelled the points on the pegboard $ABCD$. There are four possible triangles: $ABC$, $ABD$, $ACD$ and $BCD$. However these triangles are all the same shape (you can see this by rotating triangle $ABC$) so we could say that there is only one type of triangle that we can make. This triangle has angles $90^\circ$, $45^\circ$ and $45^\circ$. I know this because if you draw a square around the points $ABCD$ and cut it in along the diagonal you get this triangle. For the six point board, I again labelled the points as $ABCDEF$. There are three possible triangles <ul> <li>$ABC$, with angles $120^\circ$, $30^\circ$ and $30^\circ$.</li> <li>$ABD$, with angles $90^\circ$, $60^\circ$ and $30^\circ$.</li> <li>$ACE$, with all angles $60^\circ$ (an equilateral triangle).</li> </ul> <div>For the eight point board, I again labelled the pegs $ABCDEFGH$</div> <div>There are five possible triangles</div> <ul> <li>ABC, with angles $135^\circ$, $22.5^\circ$ and $22.5^\circ$.</li> <li>ABD, with angles $112.5^\circ$, $22.5^\circ$ and $45^\circ$.</li> <li>ABE, with angles $90^\circ$, $22.5^\circ$ and $67.5^\circ$.</li> <li>ACE, with angles $90^\circ$, $45^\circ$ and $45^\circ$.</li> <li>ACF, with angles $45^\circ$, $67.5^\circ$ and $67.5^\circ$.</li> </ul> Can you see how she worked out the angles in the triangles? If you have come across circle theorems you may find these helpful. Remember that the angles in a triangle add up to 180 degrees! You can divide the triangle (or the circle) into pieces whose angles you know to help you. If you would like to have a go at this problem for yourself, you might like to print off these sheets if you're not using the interactivity: Sheet of <a href="/content/id/2850/fourpeg.doc">four-peg</a> boards Sheet of <a href="/content/id/2850/sixpeg.doc">six-peg</a> boards Sheet of <a href="/content/id/2850/eightpeg.doc">eight-peg</a> boards </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><div class="embed"> <h2>Triangles All Around</h2> You might like to have a look at <a href="http://nrich.maths.org/public/viewer.php?obj_id=2852&part=index">Nine-Pin Triangles</a> before trying this problem. How many different triangles can you draw on a circular pegboard which has four equally spaced pegs? What are the angles of each triangle? If you have a six-peg circular pegboard, how many different triangles are possible now? What are their angles? How many different triangles could you draw on an eight-peg board? Can you find the angles of each? You may like to use the interactivity to try out your ideas. <a href="/content/id/2850/circleAngles.swf">Full Size Version</a> <mdo:flash height="400" id="/content/id/2850/circleAngles.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/2850/circleAngles.swf"></param> <param name="flashplayerversion" value="7"></param> <param name="height" value="400"></param> <param name="width" value="550"></param> </mdo:flash> For printable sets of circle templates for use with this activity, please see <a href="http://nrich.maths.org/public/viewer.php?obj_id=6676&part=index">Printable Resources page</a>. Many thanks to Geoff Faux who introduced us to the merits of the 9 pin circular geo-board. The boards, moulded in crystal clear ABS that can be used on an OHP (185 cm in diameter), together with a teacher's guide, are available from Geoff at <a href="http://www.education-initiatives.co.uk/">Education Initiatives</a> </div> This problem offers an opportunity for pupils to work in a systematic way, using their knowledge of the properties of triangles. A useful discussion about which triangles are the same and which are different could be encouraged. Students will need to apply what they know about angles in circles and triangles in order to calculate the angles in each triangle they draw. The problem encourages them to be clear about what they do know and what they can work out from it. If working on paper rather than using the interactivity, pupils may find it helpful to print these sheets off: Sheet of <a href="/content/id/2850/fourpeg.doc">four-peg</a> boards Sheet of <a href="/content/id/2850/sixpeg.doc">six-peg</a> boards Sheet of <a href="/content/id/2850/eightpeg.doc">eight-peg</a> boards </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">On four pegs, can only draw one triangle. It has angles of 90, 45, 45 (can be split into two isosceles which each have a radius as two sides). On six pegs, can draw three triangles. One has angles of 120, 30 and 30. One has angles of 60, 30 and 90. One has angles of 60, 60 and 60. On eight peg board, can make five triangles. One has angles of 135, 22.5 and 22.5. One has angles of 112.5, 45 and 22.5. One has angles of 90, 67.5 and 22.5. One has angles of 90, 45 and 45. One has angles of 67.5, 67.5 and 45. </mdoxml> 2 5 0 1 0 0 0 Triangles all Around Can you find all the different triangles on these peg boards, and find their angles? Using, Applying and Reasoning about Mathematics Working systematically Information and Communications Technology Interactivities 2D Geometry, Shape and Space Mixed triangles 2D Geometry, Shape and Space Angle properties of shapes Mathematics Tools Pinboard/geoboard