Basic High School Geometry Book
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chap 3
1) A transformation is a change in position, shape, or size of a figure. A figure can flip, slide over, turn, or change size. An image is the result of a transformation. The original figure is called a preimage. If the preimage and the image are congruent, it is an isometry. A transformation maps ( 6 ) a figure onto its image. On an image prime notation is usually used. In this case each point on the image adds a (N ) and is read as ,for example ( K N), K prime. All corresponding points are listed in the same order just as in congruent points of congruent and similar figures.
A reflection is one example of a transformation. It would be like putting a word against a mirror, which changes the orientation of the word. A reflection reverses orientation and is a form of isometry.
A translation is another example of transformation. It is the sliding of a figure to another position. It moves points the same distance and in the same direction. A translation is also a form of isometry, but does not change orientation. A composition describes any two transformations in which the second transformation is performed on the image of the image of the first transformation, for example, using reflection then translation.
A rotation, a transformation, is the turning of a figure around a point, the center of rotation. It move in either direction, either clockwise or counterclockwise, and can move at any degree. Each point simply moves in an arc around the point at angles. A rotation is isometric, but does not change orientation.
( Theorem 3-1 )Two reflections, or a composition of reflections, in two parallel lines results in translation. ( Theorem 3-2) A composition of reflections in two intersecting lines is a rotation. ( Theorem 3-3 ) In a plane, two congruent figures can be mapped onto one another by a composition of at most three reflections. Paper-folders use these statements to there advantage. If the three reflections intersect in more than one point it is called a glide reflection, another translation. Because two reflections become a translation, the figure translates, or“glides”, then reflects in a line perpendicular to that of the translation.
Any figure that can be mapped into itself by isometry has symmetry. A figure has reflectional symmetry if when it is flipped in half by any line, or the line of symmetry, it contains the same identical figure. A figure has rotational symmetry, or line symmetry, if there is a rotation of 180o or less that will map the figure into itself. If the rotation is in fact 180o, it is called a half-turn. If the half-turn maps the figure into itself it is called point symmetry.
Many factors must be known for enlargement, making a figure larger, or reduction, making a figure smaller, of a figure. The number used to multiply each point by in order to enlarge or reduce it is known as the scale factor (n){normally a, ˝, 2,3, . . .}. If n is greater than 0, the image is going to be larger, and if n is less than 0, the image will be smaller. One way to change the size of a figure is to use dilation. Place a point outside of the figure. Draw a line from the point to each point on the figure and measure the distance between each. Multiply the measures by the scale factor and place the prime notation on the line. Then simply connect the dots. This method is also known as similarity transformation.
Another method is to use a matrix, place all x-coordinates on first column and all y-coordinates on the second. Then simply multiply each number by the scale factor and graph the new points. This method is called scalar multiplication, which makes the image either form inside of the preimage or surrounding the preimage. Using similar transformation slides the enlarged or reduced image.