Lesson 4-6 BIG IDEA Matrices for Reflections Matrices can represent refl ections Vocabular refl ection image of a point over a line refl ecting line, line of refl ection refl ection What Is a Reflection? Recall from geometr that the reflection image of a point A over a line m is: 1 the point A, if A is on m; _ 2 the point A such that m is the perpendicular bisector of AA, if A is not on m The line m is called the reflecting line or line of reflection A reflection is a transformation that maps a figure to its reflection image The figure on the right is the reflection image of a drawing and the point A over the line m This transformation is called r m, and we write A = r m (A) m A A Mental Math Use matrices A, B, and C below Tell whether it is possible to calculate each of the following 2 3 A = 1 7 B = 4 05 C = 6 3 2 a A + B b AB c BA d BC Reflection over the -ais Suppose that A = _(, ) and B = (, ), as shown at the right Notice that the _ slope of AB, like the slope of the -ais, is zero, which means that AB is parallel to the -ais, and perpendicular to the -ais Also, because the -coordinates are the same and the -coordinates are opposites, the points are equidistant _ from the -ais So the -ais is the perpendicular bisector of AB This means that B = (, ) is the reflection image of A = (, ) over the -ais Reflection over the -ais can be denoted r -ais or r In this book we use r B = (, ) 0 A = (, ) You can write r : (, ) (, ) or r (, ) = (, ) Both are read the reflection over the -ais maps point (, ) onto point (, ) Matrices for Reflections 255
Chapter 4 Notice that = 1 + 0 0 + 1 = This means that there is a matri associated with r and proves the following theorem Matri for r Theorem is the matri for r Eample 1 If J = (1, 4), K = (2, 4), and L = (1, 7), fi nd the image of JKL under r Solution Represent r and JKL as matrices and multipl r JKL 1 2 1 4 4 7 = J K L 1 2 1 4 4 7 The image J K L has vertices J = ( 1, 4), K = ( 2, 4), and L = ( 1, 7) Check Use a DGS to plot JKL and its image J K L The preimage and image are graphed at the right It checks Remembering Transformation Matrices You have now seen matrices for size changes, scale changes, and one reflection, r You ma wonder: Is there a wa to generate a matri for an transformation A so that I do not have to just memorize them? One method of generating transformation matrices that works for reflections, rotations, and scale changes is to use the following two-step algorithm Step 1 Find the image of (1, 0) under A and write the coordinates in the first column of the transformation matri Step 2 Find the image of (0, 1) under A and write these coordinates in the second column For eample, here is a wa to remember the matri for r, the reflection over the -ais The image of (1, 0) under r is ( 1, 0) The image of (0, 1) under r is (0, 1) 256 Matrices
This general propert is called the Matri Basis Theorem Matri Basis Theorem Suppose A is a transformation represented b a 2 2 matri If A : (1, 0) ( 1, 1 ) and A : (0, 1) ( 2, 2 ), then A has the matri 1 2 1 2 Proof Let the 2 2 transformation matri for A be a b c d A : (1, 0) ( 1, 1 ) and A : (0, 1) ( 2, 2 ) Then a b c d = 1 2 1 2, and suppose Multipl the 2 2 matrices on the left side of the equation a 1 + b 0 a 0 + b 1 c 1 + d 0 c 0 + d 1 = 1 2 1 2 a b c d = 1 2 1 2 Thus, the matri for the transformation A is 1 2 1 2 Reflections over Other Lines Other transformation matrices let ou easil reflect polgons over lines other than the -ais GUIDED Eample 2 Use the Matri Basis Theorem to fi nd the transformation matri for r = Solution Find the image of (1, 0) under r = and write its coordinates in the fi rst column of the matri for r = r = (1, 0) = (?,? ) Find the image of (0, 1) under r = and write its coordinates in the second column of the matri for r = r = (0, 1) = (?,? ) is the matri for r = Matrices for Reflections 257
Chapter 4 Activit MATERIALS matri polgon application Step 1 Familiarize ourself with how a matri polgon application lets ou draw and transform polgons 1 2 1 Step 2 Enter the matri 4 4 7 which represents JKL, Step 3 Enter the transformation matri T1 = Step 4 Graph J K L, the image of JKL under the transformation represented b T1 How do the coordinates of corresponding points on J K L and JKL compare? What transformation maps JKL onto J K L? If ou are not sure, enter another polgon and transform it Step 5 Enter two more transformation matrices, T 2 = and T 3 = Repeat Step 4, answering the same questions for each of these transformations The Activit verifies that matri multiplication can be used to reflect JKL over three lines: the -ais, the line =, and the line = Graphs of the triangle and its images are shown below L = (1, 7) L = (1, 7) = L = (1, 7) K = (2, 4) J = (1, 4) J = (1, 4) K = (2, 4) K = (4, 2) J = (1, 4) K = (2, 4) J = (4, 1) L = (7, 1) L = ( 7, 1) J = ( 4, 1) J' = (1, 4) K' = (2, 4) = K = ( 4, 2) L' = (1, 7) r ( JKL) = J'K'L' r = ( JKL) = J K L r = ( JKL) = J K L 258 Matrices
In general, in mapping notation, r : (, ) (, ), r = : (, ) (, ), and r = : (, ) (, ) It is eas to prove that the matrices for r, r =, and r = are as stated in the net theorem Matrices for r, r =, and r = Theorem 1 is the matri for r 2 is the matri for r = 3 is the matri for r = Proof of 1 = 1 + 0 0 + 1 = You are asked to prove 2 and 3 in Questions 14 and 15 QY You have seen that matri multiplication can be used to perform geometric transformations such as size changes, scale changes, and reflections It is important to note how reflections, size changes, and scale changes differ All reflections preserve shape and size, so reflection images are alwas congruent to their preimages All sizechange images are similar to their preimages, but onl S 1 and S 1 ield congruent images In general, scale-change images are neither congruent nor similar to their preimages QY Let A = (1, 3), B = (4, 5), and C = ( 2, 6) Use matri multiplication to fi nd the image of ABC under r Questions COVERING THE IDEAS 1 Consider the photograph of a duck and its reflection image shown at the right For the preimage, assume that the coordinates of the piel at the tip of the bill are ( 30, 150) What are the coordinates of the piel at the tip of the bill of the image if the reflecting line is the -ais? 2 Sketch DEF with D = (0, 0), E = (0, 3) and F = (4, 0) Sketch its reflection image D'E'F' over the -ais on the same grid Matrices for Reflections 259
Chapter 4 3 Fill in the Blank Suppose that A' is the reflection image of A over m, and A' is not on line m Then m is the perpendicular bisector of? 4 What is the reflection image of a point C over a line m if C is on m? 5 Write in smbols in two was: The reflection over the -ais maps point (, ) onto point (, ) 6 Translate r = (, ) = (, ) into words Multiple Choice In 7 9, choose the matri that corresponds to the given reflection A B C D E 7 r = 8 r 9 r 10 a Write a matri for quadrilateral RUTH shown at the right b Use matri multiplication to draw R'U'T'H', its reflection image over the line with equation = True or False In 11 and 12, if the statement is false, provide a countereample 11 Reflection images are alwas congruent to their preimages 12 Reflection images are alwas similar to their preimages 13 Fill in the Blanks The matri equation 4 6 = 6 4 shows that the reflection image of the point? over the line with equation? is the point? R = (1, 0) H = (3, 5) T = (6, 2) U = (4, 1) APPLYING THE MATHEMATICS 14 Prove that equation = is a matri for the reflection over the line with 15 Prove that equation = is a matri for the reflection over the line with 16 Suppose P = (, ) and Q = (, ) Let R = (a, a) be an point on the line with equation = a Verif that PR = QR (Hint: Use the Pthagorean Distance Formula) b Fill in the Blank Therefore, the line with equation = is the? of PQ _ c What does our answer to Part b mean in terms of reflections? P = (, ) = R = (a, a) Q = (, ) 260 Matrices
REVIEW 17 a What is the image of (2, 3) under S 05, 2? b Write a matri for S 05, 2 c Describe S 05, 2 in words (Lesson 4-5) 18 When does the scale change matri a 0 represent a size 0 b change? (Lessons 4-5, 4-4) 19 Let P = 2 4, and Q = (Lesson 4-4) 3 4 a Compute the distance between P and Q b Compute the distance between P and Q if 1_ P = 2 0 1 P, and Q = 2 0 _ Q 2 2 c Does the transformation given b the matri distance? 1_ 2 0 _ 2 20 a Write the first 6 terms of the sequence defined b { c 1 = 1 c n = ( 1) n (Lesson 3-6) c n -1-3, for integers n 2 b Rewrite the second line of this recursive definition if the previous term is called c n preserve 21 The surface area A of a bo with length l, height h, and width w is given b the equation A = 2lh + 2lw + 2hw (Lesson 1-7) a Rewrite this equation if all edges are the same length, b Solve the equation in Part a for 22 On the television show The Price is Right, contestants spinning the Big Wheel have to make sure it turns at least one full revolution Describe the range in degrees d that the Wheel must turn in order for the spin to count (Previous course) EXPLORATION 23 a Either b graphing manuall, or b using a DGS, find the matri for the reflection r = 2 b To check this matri, test at least five points Make sure that at least two of the points are on the line of reflection QY ANSWER 1 4 2 3 5 6 = 1 4 2 3 5 6 ; A = (1, 3), B = (4, 5), and C = ( 2, 6) Matrices for Reflections 261