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Five-Minute Check (over Lesson 4–3) Then/Now New Vocabulary Example 1: Translation Example 2: Standardized Test Example Example 3: Dilation Key Concept: Reflection Matrices Example 4: Reflection Key Concept: Rotation Matrices Example 5: Rotation Lesson Menu

A. B. C. D. not possible A B C D 5-Minute Check 1

A. B. C. D. not possible A B C D 5-Minute Check 2

A. B. C. D. not true A B C D 5-Minute Check 3

A. B. C. D. not true A B C D 5-Minute Check 4

Matrix M is a 2 × 3 matrix. Matrix N is a 3 × 4 matrix Matrix M is a 2 × 3 matrix. Matrix N is a 3 × 4 matrix. What are the dimensions of M ● N? A. 3 × 4 B. 3 × 3 C. 2 × 4 D. 2 × 2 A B C D 5-Minute Check 5

You added, subtracted, and multiplied matrices. (Lesson 4–2 and 4–3) Use matrices for translations and dilations. Use matrices for reflections and rotations. Then/Now

vertex matrix reflection rotation coordinate matrix transformation preimage image translation dilation Vocabulary

Write the vertex matrix for quadrilateral ABCD. Translation Determine 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 translated 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. Example 1

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

Vertex Matrix Translation Vertex 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, shape and orientation. Example 1

Answer: A(–2, 3), B(1, 3), C(2, 0), D(0, –1) Translation Answer: A(–2, 3), B(1, 3), C(2, 0), D(0, –1) Example 1

What are 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? A. H(0, 5), I(1, 1), J(–3, –1), K(–4, 7) B. H(4, 1), I(5, –3), J(1, –5), K(0, 3) C. H(4, 5), I(5, 1), J(1, –1), K(0, 7) D. H(0, 1), I(1, –3), J(–3, –5), K(–4, 3) A B C D Example 1

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 G. A. (1, 0) B. (1, –4) C. (7, 0) D. (7, –4) Example 2

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

Solve the Test Item Step 1 Write a matrix equation. Let (c, d) represent the coordinates of G. Example 2

Step 2 Since these two matrices are equal, corresponding elements are equal. Solve an equation for x. –2 + x = –5 x = –3 Solve an equation for y. 2 + y = 0 y = –2 Example 2

Step 3 Use the values for x and y to find the variables for G(c, d). 4 + x = c 4 + (–3) = c 1 = c –2 + y = d –2 + (–2) = d –4 = d Answer: So, the coordinates of G are (1, –4), and the answer is B. Example 2

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. What are the coordinates of A? A. (–13, 10) B. (5, 10) C. (5, 0) D. (–13, 0) A B C D Example 2

Dilation ΔXYZ has vertices X(1, 2), Y(3, –1), and Z(–1, –2). Dilate ΔXYZ so that its perimeter is twice the original perimeter. Find the coordinates of the vertices of ΔXYZ. Then graph ΔXYZ and ΔX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. Example 3

Dilation The coordinates of the vertices of ΔXYZ are X(2, 4), Y(6, –2), and Z(–2, –4). Graph ΔXYZ and ΔXYZ. ΔXYZ has sides that are twice the length of those of ΔXYZ. Answer: The coordinates of the vertices of ΔXYZ are X(2, 4), Y(6, –2), and Z(–2, –4). The preimage and image are similar. Both figures have the same shape. Example 3

ΔABC has vertices A(2, 1), B(–3, –2), and C(1, 4) ΔABC has vertices A(2, 1), B(–3, –2), and C(1, 4). Dilate ΔABC so that its perimeter is four times the original perimeter. What are the coordinates of the vertices of ΔABC? A. B. A(8, 4), B(–12, –8), C(4, 16) C. A(8, 1), B(–12, –2), C(4, 4) D. A(2, 4), B(–3, –8), C(1, 16) A B C D Example 3

Concept

Reflection Determine 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 y-axis. Then graph the preimage and image. Write the ordered pairs as a vertex matrix. Then multiply the vertex matrix by the reflection matrix for the y-axis. Example 4

Reflection 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. Example 4

What are 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? A. P(–5, 0), E(–3, –3), N(1, –2), T(1, 1), A(–4, 2) B. P(5, 0), E(3, 3), N(–1, 2), T(–1, –1), A(4, –2) C. P(0, –5), E(3, –3), N(2, 1), T(–1, 1), A(–2, –4) D. P(5, 0), E(3, –3), N(–1, –2), T(–1, 1), A(4, 2) A B C D Example 4

Concept

Rotation Find the coordinates of the vertices of the image of ΔDEF 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. Example 5

Rotation 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. Example 5

What are the coordinates of the vertices of the image of ΔTRI with T(–1, 2), R(–3, 0), and I(–2, –2) after it is rotated 180 counterclockwise about the origin? A. T (1, –2), R(3, 0), I(2, 2) B. T (2, –1), R(0, –3), I(–2, –2) C. T (–1, 1), R(–3, 3), I(–2, 2) D. T (–1, –2), R(–3, 0), I(–2, 2) A B C D Example 5

End of the Lesson