Cyclic Quadrilaterals

6624 /www/nrich/html/content/id/6624/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <script type="text/javascript"> // // // $(document).ready(function() { $('#hideShow1').click(function(event) { event.preventDefault(); $('#hidden1').toggle(); }); $('#hideShow2').click(function(event) { event.preventDefault(); $('#hidden2').toggle(); }); $('#hideShow3').click(function(event) { event.preventDefault(); $('#hidden3').toggle(); }); $('#hideShow4').click(function(event) { event.preventDefault(); $('#hidden4').toggle(); }); $('#hideShow5').click(function(event) { event.preventDefault(); $('#hidden5').toggle(); }); }); // // // </script> This problem is in two parts. The first part consists of four similar challenges which provide building blocks to help you to solve the final challenge. You could work on them with others, but if you are working on your own, you may not need to attempt all four. Of course, you are welcome to go straight to the Final Challenge! In this problem, you will be working with cyclic quadrilaterals. A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. There is an interactivity at the bottom of the page which you can use to create cyclic quadrilaterals. Click on one of the questions below to get started. <a id="hideShow1" href="#">Question A</a> <div style="display: none;" id="hidden1">Using a circle with nine evenly spaced points around the edge, draw triangles which use the centre of the circle as one of the vertices. You may find these <a href="/content/id/6624/9dot.pdf">nine-point circles</a> useful. How many different triangles can you find? Work out all the angles in these triangles. On a nine-point circle, we can join four points to create cyclic quadrilaterals. Here are two examples: <mdo:image width="483" height="245" alt="" src="quads.jpg"></mdo:image> Create at least six different cyclic quadrilaterals on nine-point circles. Split your quadrilaterals into triangles, and use your earlier results to work out the angles in your quadrilaterals. What do you notice about the angles on opposite vertices of your quadrilaterals? </div> <a id="hideShow2" href="#">Question B</a> <div style="display: none;" id="hidden2">Using a circle with twelve evenly spaced points around the edge, draw triangles which use the centre of the circle as one of the vertices. You may find these <a href="/content/id/6624/12dot.pdf">twelve-point circles</a> useful. How many different triangles can you find? Work out all the angles in these triangles. On a twelve-point circle, we can join four points to create cyclic quadrilaterals. Here are two examples: <mdo:image width="483" height="231" alt="" src="cyclic12.png"></mdo:image> Create at least six different cyclic quadrilaterals on twelve-point circles. Split your quadrilaterals into triangles, and use your earlier results to work out the angles in your quadrilaterals. What do you notice about the angles on opposite vertices of your quadrilaterals? </div> <a id="hideShow3" href="#">Question C</a> <div style="display: none;" id="hidden3">Using a circle with fifteen evenly spaced points around the edge, draw triangles which use the centre of the circle as one of the vertices. You may find these <a href="/content/id/6624/15dot.pdf">fifteen-point circles</a> useful. How many different triangles can you find? Work out all the angles in these triangles. On a fifteen-point circle, we can join four points to create cyclic quadrilaterals. Here are two examples: <mdo:image width="483" height="231" alt="" src="cyclic15.png"></mdo:image> Create at least six different cyclic quadrilaterals on fifteen-point circles. Split your quadrilaterals into triangles, and use your earlier results to work out the angles in your quadrilaterals. What do you notice about the angles on opposite vertices of your quadrilaterals? </div> <a id="hideShow4" href="#">Question D</a> <div style="display: none;" id="hidden4">Using a circle with eighteen evenly spaced points around the edge, draw triangles which use the centre of the circle as one of the vertices. You may find these <a href="/content/id/6624/18dot.pdf.pdf">eighteen-point circles</a> useful. How many different triangles can you find? Work out all the angles in these triangles. On an eighteen-point circle, we can join four points to create cyclic quadrilaterals. Here are two examples: <mdo:image width="483" height="231" alt="" src="cyclic18.png"></mdo:image> Create at least six different cyclic quadrilaterals on eighteen-point circles. Split your quadrilaterals into triangles, and use your earlier results to work out the angles in your quadrilaterals. What do you notice about the angles on opposite vertices of your quadrilaterals? </div> <a id="hideShow5" href="#">Final Challenge</a> <div style="display: none;" id="hidden5">What can you say about the angles on opposite vertices of any cyclic quadrilateral? Use a diagram like the one below to create a convincing argument. <mdo:image width="243" height="231" src="cyclic.png" alt=""></mdo:image> Does your argument still work if the centre of the circle is not within the quadrilateral? <mdo:image width="243" height="231" src="cyclic2.png" alt=""></mdo:image> </div> You may find this Virtual Geoboard useful. <a href="/content/id/6624/circleAngles.swf">Full Screen Version</a> <mdo:flash height="400" width="550"><param value="/content/id/6624/circleAngles.swf" name="movie" ></param><param value="7" name="flashplayerversion" ></param></mdo:flash> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This problem invited students to consider the relationship between opposite angles of cyclic quadrilaterals. Often, with these types of problems, it is helpful to draw diagrams; several students submitted diagrams as part of their solution - well done. The problem is divided into two parts: the first part contains questions that form "building blocks" to help meet the final challenge in the second part. Nick, from St Stephen's at Carramar summed up his solution: The sum of the angles at opposite vertices of a cyclic quadrilateral is $180^\circ$. This is the same for all cyclic quadrilaterals, regardless of the positioning of the centre dot. Click <a href="/content/id/6624/izvGBz-Cyclic%20Quadrilaterals.docx" class="editorial">here</a> to see his full solution with diagrams. Well done also to the following students, who also submitted similar (and correct!) answers to this problem: Andre, Laura, Sascha, Chris and Sailesh from St. Stephen's School, Marcus and Kye from St Philip's Primary School, and Natasha. Now that you have completed this problem, you could try the following problems as an extension: <a href="http://nrich.maths.org/2845&part=Subtended Angles" class="editorial">Subtended Angles</a> and/or <a href="http://nrich.maths.org/2847&part=Right Angles" class="editorial">Right Angles</a>. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <div>This problem involves a significant 'final challenge' which can either be tackled on its own or after working on a set of related 'building blocks' designed to lead students to helpful insights. It is well suited for students who are working on circle theorems, or for applying basic understanding of angles in triangles.</div> <div>Initially working on the building blocks gives students the opportunity to then work on harder mathematical challenges than they might otherwise attempt.</div> <div>The problem is structured in a way that makes it ideal for students to work on in small groups.</div> <h3>Possible approach</h3> <div>Hand out a set of building block cards (<a href="/content/id/6624/CyclicQuads.doc">Word</a>, <a href="/content/id/6624/CyclicQuads.pdf">PDF</a>) to each group of three or four students. (The final challenge will need to be removed to be handed out later.)</div> <div>Each student, or pair of students, could be given their own building block to work on. After they have had an opportunity to make progress on their question, encourage them to share their findings with each other and work together on each other's tasks. As the four introductory tasks are very similar, anything that one student finds useful can be shared with the rest of the group to help them to make progress on their own building block.</div> <div>When everyone in the group is satisfied that they have explored the challenges in the building blocks, encourage them to discuss the similarities between their findings. Before giving the group the final challenge, ask if they can predict what they will be asked to do. Then set them the final challenge.</div> <div>The teacher's role is to challenge groups to explain and justify their mathematical thinking, so that all members of the group are in a position to contribute to the solution of the challenge.</div> <div>It is important to set aside some time at the end for students to share and compare their findings and explanations, whether through discussion or by providing a written record of what they did.</div> <h3>Key questions</h3> <div>What important mathematical insights does my building block give me?</div> <div>How can these insights help the group tackle the final challenge?</div> <div>What would happen if I tried the same process with 10- 11- 13- or 200-point circles?</div> <h3>Possible extension</h3> Other circle theorems can be explored in a similar way in the problems <a href="http://nrich.maths.org/2845&part=">Subtended Angles</a> and <a href="http://nrich.maths.org/2847&part=">Right Angles</a>. <h3>Possible support</h3> <div>Encourage groups not to move on until everyone in the group understands. The building blocks could be distributed within groups in a way that plays to the strengths of particular students.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Draw lines from the centre point to the vertices of your quadrilaterals in order to construct some isosceles triangles: <mdo:image height="231" width="243" alt="" src="cyclichint1.png"></mdo:image><mdo:image height="231" width="243" alt="" src="cyclichint2.png"></mdo:image> </mdoxml> 2 3 0 0 1 0 0 Cyclic Quadrilaterals What can you say about the angles on opposite vertices of any cyclic quadrilateral? Working on the building blocks will give you insights that may help you to explain what is special about them. 2D Geometry, Shape and Space Angle properties of shapes 2D Geometry, Shape and Space Circle theorems Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof