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Five-Minute Check (over Chapter 8) NGSSS Then/Now New Vocabulary Key Concept: Reflection in a Line Example 1: Reflect a Figure in a Line Example 2: Real-World Example: Minimize Distance by Using a Reflection Example 3: Reflect a Figure in a Horizontal or Vertical Line Key Concept: Reflection in the x- or y-axis Example 4: Reflect a Figure in the x- or y-axis Key Concept: Reflection in Line y = x Example 5: Reflect a Figure in the Line y = x Concept Summary: Reflection in the Coordinate Plane Lesson Menu

Find the geometric mean between 8 and 15. State the exact answer. 5-Minute Check 1

Determine whether the numbers 6, 9, and 12 are the measures of the sides of a right triangle. A. yes B. no A B 5-Minute Check 2

A B C D Find cos A for ΔABC if mC = 90, AB = x, AC = y, and CB = z. 5-Minute Check 3

A B C D Find sin A for ΔABC if mC = 90, AB = x, AC = y, and CB = z. 5-Minute Check 4

A B C D Find tan B for ΔABC if mC = 90, AB = x, AC = y, and CB = z. 5-Minute Check 5

A B C D If mA = 42, AB = 7, and BC = 6 in ΔABC, what is mB? A. 26.5 5-Minute Check 6

MA.912.G.2.4 Apply transformations to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons. MA.912.G.2.6 Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane. Also addresses MA.912.G.8.6. NGSSS

Draw reflections in the coordinate plane. You identified reflections and verified them as congruence transformations. (Lesson 4–7) Draw reflections. Draw reflections in the coordinate plane. Then/Now

line of reflection Vocabulary

Concept

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: 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

A B C D Draw the reflected image of quadrilateral ABCD in line n. Example 1

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

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

MINIATURE GOLF Omar is playing miniature golf at a local course 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 B C D 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 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

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. 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. 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

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

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) A B C D 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) A B C D 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

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) A B C D Example 5

Concept

End of the Lesson