Mathematical Practices 5 Use appropriate tools strategically. Content Standards G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Mathematical Practices 5 Use appropriate tools strategically. 7 Look for and make use of structure. CCSS

You identified reflections and verified them as using basic language. SWBAT use mathematical vocabulary (key terms in geometry) IOT describe transformations in the coordinate plane. SWBAT graph pre-images and images under reflection and rotation using a variety of tools IOT analyze the properties of transformations. Then/Now

Concept

Minimize Distance by Using a Reflection BILLIARDS Suppose that you must bounce the cue ball off side A before it rolls into the pocket at B. Locate the point C along side A that the ball must hit to ensure that it will roll directly toward the pocket. Example 2

Minimize Distance by Using a Reflection Understand You are asked to locate a point C on side A such that the cue ball will bounce off to roll into the pocket at point B. Plan For the cue ball to roll into the pocket at B, it must hit point C on side A somewhere in between where it sits now and the pocket at B. Use the reflection of point B on the continuation of the line that B lies on to help find this point. Example 2

Minimize Distance by Using a Reflection Solve Draw point such that the corner pocket is the midpoint between B and B'. Connect the cue ball with B' using a line. Locate point C at the intersection of the line drawn and side A. Example 2

Minimize Distance by Using a Reflection Answer: Example 2

Check Check that BC  B'C so that ΔBCB' is an isosceles triangle. Minimize Distance by Using a Reflection Answer: Check Check that BC  B'C so that ΔBCB' is an isosceles triangle. Example 2

Draw the reflected image of quadrilateral WXYZ in line p. Reflect a Figure in a Line Draw the reflected image of quadrilateral WXYZ in line p. Step 1 Draw segments perpendicular to line p from each point W, X, Y, and Z. Step 2 Locate W', X', Y', and Z' so that line p is the perpendicular bisector of Points W', X', Y', and Z' are the respective images of W, X, Y, and Z. Example 1

Step 3 Connect vertices W', X', Y', and Z'. Reflect a Figure in a Line Step 3 Connect vertices W', X', Y', and Z'. Answer: Example 1

Step 3 Connect vertices W', X', Y', and Z'. Reflect a Figure in a Line Step 3 Connect vertices W', X', Y', and Z'. Answer: Since points W', X', Y', and Z' are the images of points W, X, Y, and Z under reflection in line p, then quadrilateral W'X'Y'Z' is the reflection of quadrilateral WXYZ in line p. Example 1

Draw the reflected image of quadrilateral ABCD in line n. Example 1

Draw the reflected image of quadrilateral ABCD in line n. Example 1

A. Determine how far the obstructing wall is from the ball. MINIATURE GOLF Omar is playing miniature golf at a local course. Because a wall is blocking his direct shot, he needs to bounce the ball off wall W and hit the hole located at point H. Which of these steps would be needed to determine where on wall W Omar should aim? A. Determine how far the obstructing wall is from the ball. B. Reflect point H over the line formed by wall W. C. Determine the exact length of wall W. D. Find the perpendicular distance from the hole to the wall. Example 2

A. Determine how far the obstructing wall is from the ball. MINIATURE GOLF Omar is playing miniature golf at a local course. Because a wall is blocking his direct shot, he needs to bounce the ball off wall W and hit the hole located at point H. Which of these steps would be needed to determine where on wall W Omar should aim? A. Determine how far the obstructing wall is from the ball. B. Reflect point H over the line formed by wall W. C. Determine the exact length of wall W. D. Find the perpendicular distance from the hole to the wall. Example 2

Reflect a Figure in a Horizontal or Vertical Line A. Quadrilateral JKLM has vertices J(2, 3), K(3, 2), L(2, –1), and M(0, 1). Graph JKLM and its image over x = 1. Example 3

Reflect a Figure in a Horizontal or Vertical Line Use the horizontal grid lines to find a corresponding point for each vertex so that each vertex and its image are equidistant from the line x = 1. Answer: Example 3

Reflect a Figure in a Horizontal or Vertical Line Use the horizontal grid lines to find a corresponding point for each vertex so that each vertex and its image are equidistant from the line x = 1. Answer: Example 3

Reflect a Figure in a Horizontal or Vertical Line B. Quadrilateral JKLM has vertices J(2, 3), K(3, 2), L(2, –1), and M(0, 1). Graph JKLM and its image over y = –2. Example 3

Reflect a Figure in a Horizontal or Vertical Line Use the vertical grid lines to find a corresponding point for each vertex so that each vertex and its image are equidistant from the line y = –2. Answer: Example 3

Reflect a Figure in a Horizontal or Vertical Line Use the vertical grid lines to find a corresponding point for each vertex so that each vertex and its image are equidistant from the line y = –2. Answer: Example 3

A. Quadrilateral ABCD has vertices A(1, 2), B(0, 1), C(1, –2), and D(3, 0). Graph ABCD and its image over x = 2. A. B. C. D. Example 3

A. Quadrilateral ABCD has vertices A(1, 2), B(0, 1), C(1, –2), and D(3, 0). Graph ABCD and its image over x = 2. A. B. C. D. Example 3

B. Quadrilateral WXYZ has vertices W(2, 4), X(3, 3), Y(2, 0), and Z(0, 2). Graph WXYZ and its image over y = –1. A. B. C. D. Example 3

B. Quadrilateral WXYZ has vertices W(2, 4), X(3, 3), Y(2, 0), and Z(0, 2). Graph WXYZ and its image over y = –1. A. B. C. D. Example 3

Concept

Multiply the y-coordinate of each vertex by –1. (x, y) → (x, –y) Reflect a Figure in the x- or y-axis A. Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3) and its image reflected in the x-axis. Multiply the y-coordinate of each vertex by –1. (x, y) → (x, –y) A(1, 1) → A'(1, –1) B(3, 2) → B'(3, –2) C(4, –1) → C'(4, 1) D(2, –3) → D'(2, 3) Example 4

Reflect a Figure in the x- or y-axis Answer: Example 4

Reflect a Figure in the x- or y-axis Answer: Example 4

Multiply the x-coordinate of each vertex by –1. (x, y) → (–x, y) Reflect a Figure in the x- or y-axis B. Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3) and its reflected image in the y-axis. Multiply the x-coordinate of each vertex by –1. (x, y) → (–x, y) A(1, 1) → A'(–1, 1) B(3, 2) → B'(–3, 2) C(4, –1) → C'(–4, –1) D(2, –3) → D'(–2, –3) Example 4

Reflect a Figure in the x- or y-axis Answer: Example 4

Reflect a Figure in the x- or y-axis Answer: Example 4

A. Graph quadrilateral LMNO with vertices L(3, 1), M(5, 2), N(6, –1), and O(4, –3) and its reflected image in the x-axis. Select the correct coordinates for the new quadrilateral L'M'N'O'. A. L'(3, –1), M'(5, –2), N'(6, 1), O'(4, 3) B. L'(–3, 1), M'(–5, 2), N'(–6, –1), O'(–4, –3) C. L'(–3, –1), M'(–5, –2), N'(–6, 1), O'(–4, 3) D. L'(1, 3), M'(2, 5), N'(–1, 6), O'(–3, 4) Example 4

A. Graph quadrilateral LMNO with vertices L(3, 1), M(5, 2), N(6, –1), and O(4, –3) and its reflected image in the x-axis. Select the correct coordinates for the new quadrilateral L'M'N'O'. A. L'(3, –1), M'(5, –2), N'(6, 1), O'(4, 3) B. L'(–3, 1), M'(–5, 2), N'(–6, –1), O'(–4, –3) C. L'(–3, –1), M'(–5, –2), N'(–6, 1), O'(–4, 3) D. L'(1, 3), M'(2, 5), N'(–1, 6), O'(–3, 4) Example 4

B. Graph quadrilateral LMNO with vertices L(–1, 0), M(1, 1), N(2, –2), and O(0, –4) and its reflected image under the y-axis. Select the correct coordinates for the point M' in the new quadrilateral L'M'N'O'. A. L'(–1, 0), M'(1, –1), N'(2, 2), O'(0, 4) B. L'(1, 0), M'(–1, 1), N'(–2, –2), O'(0, –4) C. L'(1, 0), M'(–1, –1), N'(–2, 2), O'(0, 4) D. L'(0, –1), M'(1, 1), N'(–2, 2), O'(–4, 0) Example 4

B. Graph quadrilateral LMNO with vertices L(–1, 0), M(1, 1), N(2, –2), and O(0, –4) and its reflected image under the y-axis. Select the correct coordinates for the point M' in the new quadrilateral L'M'N'O'. A. L'(–1, 0), M'(1, –1), N'(2, 2), O'(0, 4) B. L'(1, 0), M'(–1, 1), N'(–2, –2), O'(0, –4) C. L'(1, 0), M'(–1, –1), N'(–2, 2), O'(0, 4) D. L'(0, –1), M'(1, 1), N'(–2, 2), O'(–4, 0) Example 4

Concept

Interchange the x- and y-coordinates of each vertex. (x, y) → (y, x) Reflect a Figure in the Line y = x Quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under reflection of the line y = x. Interchange the x- and y-coordinates of each vertex. (x, y) → (y, x) A(1, 1) → A'(1, 1) B(3, 2) → B'(2, 3) C(4, –1) → C'(–1, 4) D(2, –3) → D'(–3, 2) Example 5

Reflect a Figure in the Line y = x Answer: Example 5

Reflect a Figure in the Line y = x Answer: Example 5

Quadrilateral EFGH has vertices E(–3, 1), F(–1, 3), G(1, 2), and H(–3, –1). Graph EFGH and its image under reflection of the line y = x. Select the correct coordinates for the point H' in the new quadrilateral E'F'G'H'. A. E'(–3, –1), F'(–1, –3), G'(1, –2), H'(–3, 1) B. E'(3, –1), F'(1, –3), G'(–1, 2), H'(3, –1) C. E'(1, –3), F'(3, –1), G'(2, 1), H'(–1, –3) D. E'(–1, 3), F'(–3, 1), G'(–2, –1), H'(1, 3) Example 5

Quadrilateral EFGH has vertices E(–3, 1), F(–1, 3), G(1, 2), and H(–3, –1). Graph EFGH and its image under reflection of the line y = x. Select the correct coordinates for the point H' in the new quadrilateral E'F'G'H'. A. E'(–3, –1), F'(–1, –3), G'(1, –2), H'(–3, 1) B. E'(3, –1), F'(1, –3), G'(–1, 2), H'(3, –1) C. E'(1, –3), F'(3, –1), G'(2, 1), H'(–1, –3) D. E'(–1, 3), F'(–3, 1), G'(–2, –1), H'(1, 3) Example 5

Concept