TRANSFORMATIONS. The original figure is called the pre-image; the new (copied) picture is called the image of the transformation.

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1 Quiz Review Sheet A transformation is a correspondence that maps a point. TRANSFORMATIONS The original figure is called the pre-image; the new (copied) picture is called the image of the transformation. Types of Transformation 1. Reflections (flipping/ mirror image) a. Point reflection b. Line reflection c. Line and point symmetry 2. Translation (move across, slide) 3. Dilations (stretched or shrunk) 4. Rotations (turned) REFLECTION A point reflection exists when a figure is built around a single point called the center of the figure, or point of reflection. For every point in the figure, there is another point found directly opposite it on the other side of the center such that the point of reflection becomes the midpoint of the segment joining the point with its image. Under a point reflection, figures do not change size. Reflection Through the Origin Reflect (a, b) through the origin, then it is reflected to the third quadrant point ( a, b). The distance from the origin to (a, b) is equal to the distance from the origin to ( a, b). : RULE A reflection over a line is a transformation in which each point of the original figure (pre-image) has an image that is the same distance from the line of reflection as the original point but is on the opposite side of the line. Remember that a reflection is a flip. Under a reflection, the figure does not change size. Reflection across x-axis When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite. The reflection of the point (x, y) across the x-axis is the point (x, -y). :RULE

2 Reflecting over the y-axis When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite. The reflection of the point (x, y) across the y-axis is the point (-x, y). :RULE Reflecting over the line y = x or y = -x When you reflect a point across the line y = x, the x-coordinate and the y- coordinate change places. When you reflect a point across the line y = -x, the x- coordinate and the y-coordinate change places and are negated (the signs are changed). The reflection of the point (x, y) across the line y = x is the point (y,x). The reflection of the point (x, y) across the line y = -x is the point (-y, -x). :RULE for y = x :RULE for y = -x Reflecting over any line Each point of a reflected image is the same distance from the line of reflection as the corresponding point of the original figure. In other words, the line of reflection lies directly in the middle between the figure and its image -- it is the perpendicular bisector of the segment joining any point to its image. DILATION A dilation is a transformation (notation ) that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure. A dilation is a transformation of the plane,, such that if O is a fixed point, k is a non-zero real number, and P' is the image of point P, then O, P and P' are collinear and Rule:

3 ROTATION A rotation is a transformation that turns a figure about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. An object and its rotation are the same shape and size, but the figures may be turned in different directions. A rotation is an isometry where if P is a fixed point in the plane, is any angle and then where and. A rotation turns a figure through an angle about a fixed point called the center. When working in the coordinate plane, assume the center of rotation to be the origin unless told otherwise. A positive angle of rotation turns the figure counterclockwise, and a negative angle of rotation turns the figure in a clockwise direction. Rotation of 90 : Rotation of 180 : (same as point reflection in origin) Rotation of 270 : TRANSLATIONS A translation "slides" an object a fixed distance in a given direction. The original object and its translation have the same shape and size, and they face in the same direction. A translation creates a figure that is congruent with the original figure and preserves distance (length) and orientation (lettering order). A translation (notation ) is a transformation of the plane that slides every point of a figure the same distance in the same direction. : RULE

4 COMPOSITIONS When two or more transformations are combined to form a new transformation, the result is called a composition of transformations. In a composition, the first transformation produces an image upon which the second transformation is then performed. The symbol for a composition of transformations is an open circle. The notation is read as a reflection in the x- axis following a translation of (x+3, y+4). **Be careful!!! The process is done in reverse!! 2 nd 1 st GLIDE REFLECTIONS When a translation (a slide or glide) and a reflection are performed one after the other, a transformation called a glide reflection is produced. In a glide reflection, the line of reflection is parallel to the direction of the translation. It does not matter whether you glide first and then reflect, or reflect first and then glide. This transformation is commutative. A glide reflection is a transformation in the plane that is the composition of a line reflection and a translation through a line (a vector) parallel to that line of reflection. is the image of translation through the vector v. under a glide reflection that is a composition of a reflection over the line and a OR :RULE ISOMETRY Isometry - length is preserved - the figures are congruent. Direct Isometry - orientation is preserved - the order of the lettering in the figure and the image are the same, either both clockwise or both counterclockwise. Opposite Isometry - orientation is not preserved - the order of the lettering is reversed, either clockwise becomes counterclockwise or counterclockwise becomes clockwise.

5 Line Reflection Point Reflection Translations Rotations Dilations Opposite isometry Direct isometry Direct isometry Direct isometry NOT isometry. Figures are similar. 1. angle measure 2. parallelism 3. collinearity 4. midpoint Lengths not same. Reverse Orientation (letter order changed) Notation: Notation: Notation: Notation: Notation:

6 3D SOLIDS Theorem 12-1 The lateral area of a right prism equals the perimeter of a base times the height of the prism. (LA = ph) Theorem 12-2 The volume of a right prism equals the area of a base times the height of the prism. (V=Bh) Theorem 12-3 The lateral are of a regular pyramid equals half the perimeter of the base times the slant height. (LA = ½Bh) Theorem 12-4 The volume of a pyramid equals one third the area of the base times the height of the pyramid. (V= ⅓Bh) Theorem 12-5 The lateral area of a cylinder equals the circumference of the base times the height of the cylinder. (LA =2πrh) Theorem 12-6 The volume of a cylinder equals the area of a base times the height of the cylinder. (V = πr²h) Theorem 12-7 The lateral area of a cone equals half the circumference of the base times the slant height. (LA = πrl) Theorem 12-8 The volume of a cone equals one third the area of the base times the height of the cone. (V=⅓πr²h) Theorem 12-9 The area of a sphere equals 4π times the square of the radius. (A= 4πr²) Theorem The volume of a sphere equals π times the cube of the radius. (V = π )