 An image is the new figure, and the preimage is the original figure  Transformations-move or change a figure in some way to produce an image

A translation is an isometry. ▲ABC is congruent to ▲A’B’C’ A BC A’ B’ C’

 Vectors are another way to describe translations  Have both direction and magnitude (size)  Shown by an arrow drawn from one point to another

5 units right (horizontal component) 3 units up (vertical component) The initial point, or the starting point, of the vector is F F G The terminal point, or ending point, of the vector is G The component form of a vector combines the vertical and horizontal components. So, the component form of FG is <5, 3>

A matrix is a rectangular arrangement of numbers in rows and columns. Every number inside is called an element. The dimensions of a matrix are the numbers of rows and columns. For example, the matrix above has 2 rows and 2 columns, so the dimensions of the matrix are 2 x 2 (read “2 by 2”) The element in the second row and first column is -5. row column

To add or subtract matrices, you add or subtract corresponding elements. The matrices MUST have the same dimensions.

 You can use matrix addition to represent a tradition in the coordinate plane. The image matrix for a translation is the sum of the translation matrix and the matrix that represents the preimage.

The product of two matrices A and B is defined only when the number of columns in A is equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix A X B = AB (m by n) X (n by p) = (m by p)

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 Like a mirror, there is always a “line of reflection,” or the mirror line  A reflection in line m maps every point P’; each point is true… m P P’ 1.If P is not on m, then m is the perpendicular bisector of PP ’ m PP’ 2. If P is on m, then P=P’ (prime)

1. If (a, b) is reflected on the x-axis, then the image is (a, -b) 2. If (a, b) is reflected on the y-axis, then the image is (-a, b) 3. If (a, b) is reflected on line y=x, then the image is (b, a) 4. If (a, b) is reflected on y=-x, then the image is (-b, -a)

AA reflection is an isometry Proving the Theorem Case 1 Case 2 Case 3 Case 4 Pt P and Q are on the same side as m P P PP’ Q Q Q Q P’ Q’ P and Q are on opposite sides of m Q lies on m, and PQ is not perpendicular to m Q lies on m, and PQ is perpendicular to m

Reflection in the x-axisReflection in the y-axis

A rotation is a transformation in which a figure turns about a fixed point called the “center of rotation.” Center of rotation 90 ̊ CCW rotation

 You can rotate a figure more than 180 ̊  A figure returns to itself when rotated 360 ̊

OOften a point (a, b) is rotated CCW. The following are true: 1. Rotation of 90 ̊ : (a, b) to (-b, a) 2. Rotation of 180 ̊ : (a, b) to (-a, -b) 3. Rotation of 270 ̊ : (a, b) to (b, -a)

Using matrices you can find certain images of a polygon rotated about the origin using matrix multiplication The rotation matrix must be to the left of the polygon matrix

R o t a t i o n T h e o r e m C a s e s o f T h e o r e m 9. 3  A rotation is an isometry  To prove the rotation theorem, you need to show a rotation preserves segment length Q Q’ R R’ P Case 1 : R, Q, and P are noncollinear Q’ R’ P R Q Case 2 : R, Q, and P are collinear Q’ Q R’ RP Case 3 : P and R are the same point

 Perform more than one transformation  The composition of 2 or more isometries is an isometry

If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation. If P” is the image of P, then: 1. PP” is perpendicular to k and m, and 2. PP”=2d, where d is the distance between k and m

If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P. The angle of rotation is 2x ̊, where x ̊ is the measure of the acute or right angle formed by k and m.

 A process of multiplying each element of a matrix by a real number/scalar number  Use scalar multiplication to represent a dilation. To find an image matrix of a dilation centered at the origin, use the scale factor as the scalar number