Examples of using Pythagoras - Triangles and Polygons

The following are some examples generated from Pythagoras and its Graphic window (Arena).
When using Pythagoras software, it is possible to present dynamic examples and demonstrations, as well as to experience or explore mathematical concepts and functions.


Triangle Area

	Construction 1. Draw triangle ABC and draw a parallel to the base through C. 2. Create other triangles with the same base and equal height, by dragging the point C along the parallel. 3. For each triangle enter the expression for half the base multiplied by the height, into the table. Note that the results (areas) in all the triangles are identical. Further activities Repeat the previous procedure, but without creating new triangles.
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Angles of a Triangle

	Construction Show that the sum of angles in a triangle is 180˚. To do this draw a parallel to the base DE through the vertex A. Enter the names of the angles near the vertex A with a plus sign between them into the table and do the same with the angles of the triangle. Drag point A along the parallel and note the changes in the angles' measurements and the two sums. Verification and proof Prove the theorem using logical geometrical arguments.
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Triangles and Circles

	Circumscribed circle 1. Draw a triangle and its perpendicular bisectors. Mark the point of concurrency. 2. Change the triangle and follow the changing perpendicular bisectors. Are they still concurrent? 3. Draw the circumscribed circle. Change the triangle and explore the resulting changes. Inscribed circle 1. Draw a triangle and its angle bisectors. Mark the point of concurrency. 2. Change the triangle and follow the changing angle bisector s. 3. Draw the inscribed circle. Change the triangle and explore the resulting changes.
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Sum of Sides Distances in an Equilateral Triangle

	Construction 1. Draw an equilateral triangle ABC with altitude CD. 2. From a point F inside the triangle, draw perpendiculars to the sides. 3. Activate the table and enter expressions for the altitude (CD), the distances (FH,FE,FG) and their sum (FH+FE+FG). 4. Move the point F within the triangle and follow the resulting values. Record the values at a number of locations. 5. Change the size of the triangle and follow the results. Verification and proof Prove the result, geometrically.
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Properties of an Isosceles Trapezoid

	Construction 1. Measure the trapezoid's angles and look for connections between the angle measures. 2. Change the trapezoid by dragging its vertices one by one. Do the connections discovered previously still exist? 3. Draw the circumscribed circle of the trapezoid. Change the trapezoid again. Does an isosceles trapezoid always have a circumscribed circle? 4. Can an isosceles trapezoid have an inscribed circle? 5. Change the trapezoid and measure the sides. Which sides remain equal to each other whatever the change?
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Polygons


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