




Size transformation: Coordinates of the original shape are multiplied by the scale factor to produce an image. For example, a triangle made of (1, 3), (2, 8) and (3, 5) can be scaled up, by a scale factor of 2, (using the origin as the center) to make another triangle (2, 6), (4, 16), (6, 10) which has sides twice as long as the original, and area four times as large. In general, the area of the image will be increased by a factor of a^{2} if the scale factor is a. Other kinds of transformations are as follows: translations, reflections, rotations. Each can be discerned from the patterns of change from coordinates of preimage to coordinates of image. The coordinate rule for a size transformation of magnitude 2 centered at the origin is (x, y) → (2x, 2y). (See p. 206.)
Distance between two points: The formula for the distance between two points is developed using the Pythagorean Theorem. (See p. 166.)
Midpoint of a segment: A midpoint of a segment is the point that is the same distance from each endpoint. (Students develop this formula on p. 168.)
Multiplying the coordinate matrix of a shape by a transformation matrix creates a new matrix that represents the transformed image: For example, to create a reflection over the yaxis, we multiply the original triangle matrix by . In addition, various rotations are possible. Rotation of 90° counterclockwise about the origin and rotation of 45° counterclockwise about the origin are effected by multiplying by and . Rigid transformations (translations, reflections, rotations) leave the size and shape unaffected; distances, angle measures, slopes and areas are unchanged. For example, adding the matrix to the triangle matrix will add 2 to every x value and 3 to every y value so the triangle will move 2 right and 3 up to become the new but congruent triangle .