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ROTATION |
| Geometry | |
| 1. A DEFINITION OF ROTATION | |
| A rotation of angle G about centre O transforms point A into point A' so that OA is equal to OA' and angle AOA' is equal to G. In the Descartes window the triangle ABC is transformed into triangle A'B'C' through a rotation of angle G about centre O. The angle of rotation can be changed by writing in a new value or using the arrows. The vertices of the original triangle, as well as the centre of rotation can all be moved by dragging them with the mouse. | |
| 1.- Enter different
values for the angle of rotation and look carefully at the position of
the shape which is rotated in each case.
2.- Look carefully to see whether positive and negative angles rotate in a clockwise or anticlockwise direction. 3.- What happens when the angle of rotation is 360º? What about when it is 180º? 4.- Change triangle ABC by dragging its vertices and explain the relation between the two triangles. 5.- Move point B until it is lined up with points A and C and the three vertices are in a straight line. Rotate it through different angles. What happens to a straight line when it is rotated? Do the points stay in the same order when they are rotated? |
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| 2. symmetry through A CENTRAL POINT. CENTRE OF SYMMETRY | |
| Symmetry
through a central point is an example of rotation through 180º.
Symmetry through a central point O transforms point A
into point A' so that O
is the midpoint of segment AA'. A shape has O as its
centre of symmetry if the shape, transformed through a central point
of symmetry O fits exactly on top of itself. We are going to use Descartes to try and discover some basic shapes with a centre of symmetry. |
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| 6.- Move
the centre of rotation to the geometric centre of the rectangle and
check that the initial shape coincides with the transformation by
means of symmetry through a central point. Can we say that this
rectangle has a centre of symmetry?
7.- Click on the Init button and make a square by moving points A, B, C and D with the mouse. Then, move the centre of symmetry to the centre of the square and check that the initial shape coincides with the transformation by means of symmetry through this central point. Has it got a centre of symmetry? 8.- Click on the Init button again and construct a triangle. Then, move the centre of symmetry to see if the original triangle coincides with the transformation about a central point. Repeat the exercise with different triangles such as equilateral triangles and answer the following question: Is there a triangle which has a centre of symmetry? 9.- Repeat this investigation exercise with a rhombus and analyse which types of rhombuses have a central point of symmetry. What about parallelograms? Do they have a centre of symmetry? |
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| 3. ROTATION ON THE CARTESIAN PLANE | |
| Certain types of rotation about the origin of coordinates are easy to work out, for example: 90º, 180º and 270º or -90º. | |
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10.- In your exercise book draw the coordinates of the points A(1,1), B(-2,3), C(2,-1), D(-2,-3) and their rotation about the origin through the following angles: 90º, 180º and 270º respectively. 11.- If we have a point with coordinates (x,y), work out its new coordinates after applying a transformation of symmetry about the origin O. 12.- Work out the coordinates of a square, whose vertices are A(1,1), B(-1,1), C(-1,-1) & D(1,-1), which is rotated through 90º and draw the translation in your exercise book. What is the relation between these two squares? 13.- Repeat the exercise for a rectangle, whose vertices are A(2,1), B(-2,1), C(-2,-1) & D(2,-1), which is rotated 180º. |
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| 4. combination of rotations about the same centre | |
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The combination of two rotations about the same centre is the same as another rotation whose angle is the sum of the angles of each rotation. In the following window each of the angles of rotation that triangle ABC undergoes can be changed, in order to obtain triangle A´B´C´ after the first rotation and triangle A´´B´´C´´ after the second.
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| 14.- Make the rotations through the following angles:90º & 120º; 270º & -90º; 160º & 200º; -130º & -80º. | |
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| Miguel García Reyes | ||
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| Spanish Ministry of Education. Year 2001 | ||
Except where otherwise noted, this work is licensed under a Creative Common License