![]() ![]() The point at which we do the rotation, we'll call point P. ![]() Well, let's say the shape is a triangle with vertices A, B, and C, and we want to rotate it 90 degrees. The shape is being rotated! But how do we do this for a specific angle? With your finger firmly on that point, rotate the paper on top. Now place your finger on the rotation point. Put another paper on top of it (I like to imagine this one as being something like a transparent sheet protector, and I draw on it using a dry-erase marker) and trace the point/shape. Here's something that helps me visualize it: The "formula" for a rotation depends on the direction of the rotation. I'm sorry about the confusion with my original message above. If you want to do a clockwise rotation follow these formulas: 90 = (b, -a) 180 = (-a, -b) 270 = (-b, a) 360 = (a, b). Also this is for a counterclockwise rotation. 360 degrees doesn't change since it is a full rotation or a full circle. 180 degrees and 360 degrees are also opposites of each other. So, (-b, a) is for 90 degrees and (b, -a) is for 270. Again, let’s try this first with a drawing.The way that I remember it is that 90 degrees and 270 degrees are basically the opposite of each other. To “do” an enlargement from a given centre, we draw a continued line from the centre to each of the vertices and then mark a point “n” times further along that line, where n is our scale factor. This is easier to understand by seeing it done. ![]() Where these lines meet will be the centre, and the relative distances from the centre will give us the scale factor of the enlargement. If we need to identify the centre and scale factor of an enlargement and we know the object (the original shape) and the image, we can use a special method where we draw a continued line joining each of the corresponding vertices of the object and the image. Now let’s look at a slightly more complicated type of transformation called an enlargement. In the short exercise, exercise 15D on page 241, we will practice applying more than one geometrical transformation: They may also ask you to do several transformations one after another. Questions will often want you to identify which transformation has taken place without telling you what kind of transformation you are looking for. Let’s complete exercise 15C on pages 239 and 240 of the textbook: Let’s try identifying the translation vectors for a point that has moved and marking the image point following a translation ourselves, and then let’s try the examples below where we are dealing with a whole shape: Let’s also try identifying what vector a point or shape has been translated by. Then we can try doing it with complete shapes. Let’s try doing this first with individual points in space. We use vector notation to specify this translation, putting the horizontal translation on the top and the vertical translation on the bottom. Let’s try questions 2 to 4 from exercise 15B on pages 236 to 237 of the textbook:Ī translation always has a horizontal component which moves to the right (or the left) and a vertical component which move up (or down). From than point every part of the shape rotates by a specified angle (e.g. We must remember that there is always a specific point that is the centre of rotation. Now let’s turn our attention to rotations:Ī rotation turns a shape around a certain point. Now let’s complete exercise 15A on pages 234 to 236 of the textbook: Be careful – the mirror line won’t always be horizontal or vertical! (Although it will always be straight during the iGCSE course)! If we then join up all of our new vertices we will have the image of our original shape after a reflection. To reflect a shape in a mirror line, we need to take every vertex of the shape to the same distance on the other side of the mirror line (let’s see what this means). The specific changes that we look at are: translations, rotations, reflections and enlargements. Transformational geometry is interested in taking a shape and changing it in some way. Here we are going to look at a relatively modern part of geometry called transformational geometry. “Shape & Space” or geometry is a big subject – so much so that we can talk about different types of geometry depending what we are focusing on. ![]()
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