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9.3 Performing Reflections

9.3 Performing Reflections. Reflection: a transformation that uses a line to reflect an image. A reflection is an isometry , but its orientation changes from the preimage to the image Line of reflection is the line that acts like a mirror

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9.3 Performing Reflections

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  1. 9.3Performing Reflections • Reflection: a transformation that uses a line to reflect an image. • A reflection is an isometry, but its orientation changes from the preimage to the image • Line of reflection is the line that acts like a mirror • A reflection in a line (m) maps every point (P) in the plane to a point (P΄) so that for each point, one of the following is true: • If P is on m, then P=P΄ • P m • P΄

  2. Performing Reflections • Or, if P is not on m, then m is the ⊥ bisector PP΄ P P΄

  3. Reflect AB: across the x-axis across the y-axis across the line y=x across the line y=-x

  4. Rules for Reflections • If (a,b) is reflected in the x-axis, its image is (a,-b). • If (a,b) is reflected in the y-axis, its image is (-a,b). • If (a,b) is reflected in the line y = x, its image is (b,a). • If (a,b) is reflected in the line y = -x, its image is (-b,-a).

  5. D E F D E F 1 0 –1 0 1 3 4 1 3 4 X X 2 3 0 0 1 0 -1 2 3 0 Polygon matrix Polygon matrix Reflection matrix Reflection matrix Use to find the image of a polygon reflected in the x-axis or the y-axis. The reflection matrix must be first when multiplying Reflection Matrices Across x-axis: Across y-axis:

  6. The vertices of DEFare D(1, 2), E(3, 3), and F(4, 0). Find the reflection of DEFin the y-axis using matrix multiplication. Graph DEFand its image. D E F Multiply the polygon matrix by the matrix for a reflection in the y-axis. –1 0 1 3 4 STEP 1 X 0 1 2 3 0 Polygon matrix Reflection matrix Use matrix multiplication to reflect a polygon SOLUTION

  7. –1(1) + 0(2) –1(3) + 0(3) –1(4) + 0(0) = 0(1) + 1(2) 0(3) + 1(3) 0(4) + 1(0) D′ E′ F′ –1 –3 –4 = 2 3 0 Image matrix EXAMPLE 5 Use matrix multiplication to reflect a polygon

  8. The vertices of ABC are A(1, 3),B(5, 2), and C(2, 1). Graph the reflection of ABC described. a. In the linen : x = 3 Point Ais 2 units left of n, so its reflectionA′is 2 units right of nat (5, 3). Also, B′ is 2 units left of nat (1, 2), and C′is 1 unit right of n at (4, 1). Graph reflections in horizontal and vertical lines SOLUTION

  9. The vertices of ABC are A(1, 3),B(5, 2), and C(2, 1). Graph the reflection of ABC described. b. In the linem : y = 1 Point Ais 2 units above m, so A′ is 2 units below mat (1, –1). Also, B′ is 1 unit below mat (5, 0). Because point Cis on line m, you know that C = C′. Graph reflections in horizontal and vertical lines SOLUTION

  10. Real World: Find a minimum distance You are going to meet a friend on the beach shoreline. Where should you meet in order to minimize the distances you both have to walk.?

  11. Your house is at (-6,3) and your friend’s house is at (9,6). At what point on the shoreline (x-axis) should you meet? Shoreline

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