Further Activities

The following ideas for activities are also taken from Don Steward's website. Try them on paper first and then turn to Autograph to look at them in more depth. Click on the image to download the individual Autograph files. 

Activity 1 – The Equilateral House 
Download
1. Equilateral House.agg



Challenge: A square has an equilateral triangle constructed on each of two of its adjacent sides. The top corner of each triangle is connected to each other and to the far corner of the square. Can you prove that the resulting shape is an equilateral triangle? 

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The situation has been modelled on Autograph 
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Experiment by moving the positions of the three corners of the triangle around the page 
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Convince yourself that the triangle is equilateral by measuring the sides or the angles:
– Measure sides: click on a side and the distance will be displayed in the status bar at the bottom of the screen
– Measure angles: double click on the angle arc and place a tick by Show Label 
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Does this help you prove that the triangle is equilateral? 

Activity 2 – The Mysterious Yellow Triangle 
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2. Mysterious Yellow Triangle.agg



Challenge: Draw any right angled triangle, bisect an angle and construct the perpendicular to the hypotenuse (an altitude). Can you prove that the yellow triangle is isosceles. 

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The situation has be modelled using Autograph 
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Experiment by dragging the three corners of the right angled triangle around the page 
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Does this help you prove that the yellow triangle must be isosceles? 

Activity 3 – Isosceles Split 
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3. Isosceles Split.agg



Challenge: What kinds of triangles will split into two (noncongruent) isosceles triangles? The split must be a straight line from one corner of the triangle to an edge. Don Steward claims there are three different types of triangle for which this can be done, each with a set of angles that have a special property. Can you find them? 

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Once again the situation can be modelled using Autograph 
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To change the shape of the triangles, simply drag their corners around 
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Test out your students predictions and variations for types of triangles 
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Investigate why some work and others do not 

Activity 4 – The Special Parallelogram 
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4. The Special Parallelogram.agg



Challenge: Normally when the two base angle of a parallelogram are bisected, the resulting lines do not meet on an edge of the parallelogram. When does this happen? Can you prove it? 

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This problem in particular lends itself well to being dynamically modelled and investigated using Autograph 
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To change the shape of the parallelogram, simply drag the corners. You will notice that the sizes of the angles change as well 
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As a hint towards (one possible) solution, the intersection of the two bisectors has been marked with a point. You can create and measure the angle between these two lines as follows:
– Click on the three points that define the angle in order
– Rightclick and select Angle from the dropdown menu

