Geometric Transformations (1).ppt

To transform something is to change it. In geometry, there are specific ways to describe how a figure is changed. The transformations you will learn about include: Translation Rotation Reflection Dilation  I. TRANSLATIONS ATranslation “slides” an object a fixed...

To transform something is to change it. In geometry, there are specific ways to describe how a figure is changed. The transformations you will learn about include: Translation Rotation Reflection Dilation  I. TRANSLATIONS ATranslation “slides” an object a fixed distance in a given direction. The original objectsame and its translation shape have and size theface in the same direction. and they SLIDES. Translations are Let's examine some coordinate geometry. translations related to The example shows how each vertex moves the same distance in the same direction.  Write the Points What are the coordinates for A, B, C? A (-4,5) B (-1,1) C (-4,-1) What are the coordinates for A’, B’. C’? A’ (2,5) B’ (5,1) C’ (2,-1) How are they alike? They are similar triangles How are they different? Each vertex slides 6 units to the right Write the points What are the coordinates for A, B, C, D? A (2, B (4, 4) C4)(5, D (2, 1) What are the 2) coordinates for A’, B’, C’, D’? A’ (-5, B’ (-3, C’ (-2,1) - D’ (-5, - 1)How did the 1) 2) transformation change the points? The figure slides 7 units to the left and 3  II. ROTATIONS Arotationis a transformation that turns a figure about a fixed point same called shapeof the center and size An rotation. object andmay figures its rotation are the be turned in , but the different directions  Rotate “About Vertex” Draw the first shape with the points given Then rotate it at the vertex (both figures will still touch) the amount given and in the direction given (clockwise/counterclockwise) Give the new points to the figure  Rotate “About Origin” 90o Rotation 180o Rotation 270o Rotation (x, y) (y, -x) (x, y) (-x, -y) (x, y) (-y, x) *your original point will flip *your original point will *your original point will flip flop and your original x value remain as it is, but will flop and your original y value will become the opposite sign will become the opposite sign become the opposite signs These will only work if the figure is being rotated CLOCKWISE If it’s asking for COUNTERCLOCKWISE rotation, then 90o = 270o and 270o = 90o  Examples… Triangle DEF has vertices D(-4,4), E(-1,2), & F(-3,1). D D’ F’ What are the new E coordinates of the figure if it is rotated 90o clockwise F E’ around the origin? 90o … (x, y) (y, -x) D(-4,4) = D’(4,4) E(-1,2) = E’(2,1) F(-3,1) = F’(1,3)  III. REFLECTION  Areflection can be seen in water, in a mirror, in glass, or in a shiny surface. An object and same shape and its reflection size have the , but figures face in opposite directions the. In a mirror, for example, right and left are switched. Line reflections are FLIPS!!!  The line (where a mirror may be placed) is line of reflection called the line of reflection. The distance from a point to the line of reflection is the same as the distance from the point's image to the line of reflection. A reflection can be thought of as a "flipping" of an object over the line of reflection.  Reflections on a Coordinate Plane Over the x-axis Over the y-axis (x, y) (x, -y) (x, y) (-x, y) *multiply the y-coordinates *multiply the x-coordinates by -1 by -1 (or simply take the (or simply take the opposite #) opposite #)  What happens to points in a Reflection? Name the points of the original triangle. A (2,-3) B (5,-4) C (2,-4) Name the points of the reflected triangle. A’ B’ (5,4) C’ (2,4) What (2,3) is the line of reflection? x-axis How did the points change from the original to the reflection? The sign of y switches  IV. DILATIONS  A dilation is a transformation that dilation same that produces an image shape is the same shape as different the original, but is size. a different size. larger A dilation used to create an image enlargement larger than the original is called an smaller enlargement. A dilation used to create an image smaller than the reduction original is called a reduction. Dilations always involve a change in size. Notice how EVERY coordinate of the original triangle has been  How to find a dilation You will multiply both the x and y- coordinates for each point by the scale factor. Scale factors will be given to you. Example… A figure has vertices F(-1, 1), G(1,1), H(2,-1), and I(-1,-1). Graph the figure and the image of the figure after a dilation with a scale factor of 3. F(-1, 1) = F’(-3, 3) G(1, 1) = G’(3, 3) H(2, -1) = H’(6, -3) I(-1, -1) = I’(-3, -3)  How to find a scale factor Take the measurement of both the original image and the dilated one and set up a ratio Measurement of dilation = divide #s Measurement of original  Example… Through a microscope, the image of a grain of sand with a 0.25-mm diameter appears to have a diameter of 11.25-mm. What is the scale factor of the dilation? Diameter of dilation = 11.25 = Diameter of original 0.25 *Scale factor is 45.