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Lesson 4 Contents Example 1Translate a Figure Example 2Find a Translation Matrix Example 3Dilation Example 4Reflection Example 5Rotation

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Example 4-1a Find the coordinates of the vertices of the image of quadrilateral ABCD with A(–5, –1), B(–2, –1), C(–1, –4), and D(–3, –5) if it is moved 3 units to the right and 4 units up. Then graph ABCD and its image A B C D. Write the vertex matrix for quadrilateral ABCD.

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Example 4-1b To translate the quadrilateral 3 units to the right, add 3 to each x -coordinate. To translate the figure 4 units up, add 4 to each y -coordinate. This can be done by adding the translation matrix to the vertex matrix of ABCD.

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Example 4-1c Vertex MatrixTranslationVertex Matrix of ABCD Matrix of A'B'C'D' The coordinates of A B C D are A (–2, 3), B (1, 3), C (2, 0), D (0, –1). Graph the preimage and the image. The two graphs have the same size and shape.

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Example 4-1d Answer: A (–2, 3), B (1, 3), C (2, 0), D (0, –1)

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Example 4-1e Find the coordinates of the vertices of the image of quadrilateral HIJK with H(2, 3), I(3, –1), J(–1, –3), and K(–2, 5) if it is moved 2 units to the left and 2 units up. Then graph HIJK and its image H I J K. Answer: H (0, 5), I (1, 1), J (–3, –1), K (–4, 7)

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Example 4-2a Short-Response Test Item Rectangle E F G H is the result of a translation of the rectangle EFGH. A table of the vertices of each rectangle is shown. Find the coordinates of F and G. Rectangle EFGH E(–2, 2) E (–5, 0) F F (1, 0) G(4, –2) G H(–2, –2) H (–5, –4)

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Example 4-2b Read the Test Item You are given the coordinates of the preimage and image of points E and H. Use this information to find the translation matrix. Then you can use the translation matrix to find the coordinates of F and G.

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Example 4-2c Solve the Test Item Write a matrix equation. Let (a, b) represent the coordinates of F and let (c, d) represent the coordinates of G.

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Example 4-2d Since these two matrices are equal, corresponding elements are equal. Solve an equation for x.Solve an equation for y.

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Example 4-2e Use the values for x and y to find the variables for F(a, b) and G (c, d). Answer:So, the coordinates of F are (4, 2) and the coordinates of G are (1, –4).

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Example 4-2f Short-Response Test Item Rectangle A B C D is the result of a translation of the rectangle ABCD. A table of the vertices of each rectangle is shown. Find the coordinates of D and A. Rectangle ABCD A(–4, 5) A B(–1, 5) B (8, 0) C(–1, 0) C (8, –5) D D (5, –5) Answer:The coordinates of D are (–4, 0) and the coordinates of A are (5, 0).

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Example 4-3a XYZ has vertices X(1, 2), Y(3, –1), and Z(–1, –2). Dilate XYZ so that its perimeter is twice the original perimeter. What are the coordinates of the vertices of X Y Z ? If the perimeter of a figure is twice the original figure, then the lengths of the sides of the figure will be twice the measure of the original lengths. Multiply the vertex matrix by the scale factor of 2.

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Example 4-3b The coordinates of the vertices of X Y Z are X (2, 4), Y (6, –2), and Z (–2, –4). Graph XYZ and X Y Z. The triangles are not congruent. The image has sides that are half the length of those of the original figure. Answer: X (2, 4), Y (6, –2), Z (–2, –4)

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Example 4-3c ABC has vertices A(2, 1), B(–3, –2), and C(1, 4). Dilate ABC so that its perimeter is four times the original perimeter. What are the coordinates of the vertices of A B C ? Answer: A (8, 4), B (–12, –8), C (4, 16)

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Example 4-4a Find the coordinates of the vertices of the image of pentagon PENTA with P(–3, 1), E(0, –1), N(–1, –3), T(–3, –4), and A(–4, –1) after a reflection across the x -axis. Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by the reflection matrix for the x -axis.

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Example 4-4b Answer: The coordinates of the vertices of P E N T A are P (–3, –1), E (0, 1), N (–1, 3), T (–3, 4), and A (–4, 1). Notice that the preimage and image are congruent. Both figures have the same size and shape.

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Example 4-4c Find the coordinates of the vertices of the image of pentagon PENTA with P(–5, 0), E(–3, 3), N(1, 2), T(1, –1), and A(–4, –2) after a reflection across the line y = x. Answer: P (0, –5), E (3, –3), N (2, 1), T (–1, 1), A (–2, –4)

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Example 4-5a Find the coordinates of the vertices of the image of DEF with D(4, 3), E(1, 1), and F(2, 5) after it is rotated 90 counterclockwise about the origin. Write the ordered pairs in a vertex matrix. Then multiply the vertex matrix by the rotation matrix.

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Example 4-5b Answer: The coordinates of the vertices of triangle D E F are D (–3, 4), E (–1, 1), and F (–5, 2). The image is congruent to the preimage.

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Example 4-5c Find the coordinates of the vertices of the image of TRI with T(–1, 2), R(–3, 0), and I(–2, –2) after it is rotated 180 counterclockwise about the origin. Answer: T (1, –2), R (3, 0), I (2, 2)

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End of Lesson 4

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