Chapter 8 Transformations and Congruence
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1 Lesson 8-1 Translations Page 559 Graph ABC with vertices A(1, 2), B(3, 1), and C(3, 4). Then graph the image of the triangle after it is translated 2 units left and 1 unit up, and write the coordinates of its vertices. Graph ABC. Then move each vertex of the triangle 2 units left and 1 unit up. Use prime symbols for the vertices of the image. From the graph, the coordinates of the vertices of the image are A ( 1, 3), B (1, 2), and C (1, 5). Quadrilateral KLMN has vertices K( 2, 2), L(1, 1), M(0, 4), and N( 3, 5). It is first translated by (x + 2, y 1) and then translated by (x 3, y + 4). When a figure is translated twice, a double prime symbol is used. Determine the coordinates of quadrilateral K L M N after both translations. First translate KLMN by the first translation (x + 2, y 1). Add 2 to each x-coordinate and 1 to each y-coordinate. K( 2, 2) ( 2 + 2, 2 + ( 1)) or (0, 3) L( ) (1 + 2, 1 + ( 1)) or (3, 0) M(, 4) (0 + 2, 4 + ( 1)) or (2, 3) N( 3, ) ( 3 + 2, 5 + ( 1)) or ( 1, 4) Next translate K L M N by the next translation (x 3, y + 4). Add 3 to each x-coordinate and 4 to each y-coordinate. K ' 0, 3 0 3, 3 4 or 3,1 ' 2,3 2 3,3 4 or 1,7 ' 1,4 1 3,4 4 or 4,8 L' 3,0 3 3,0 4 or 0,4 M N The coordinates after both translations are K ( 3, 1), L (0, 4), M ( 1, 7), and N ( 4, 8).
2 Lesson 8-2 Reflections Page 567 Graph GHJ with vertices at G(4, 2), H(3, 4), and J(1, 1) and its reflection over the y-axis. Write an algebraic representation that explains the effect of the reflection. Then determine the coordinates of the reflected image. The y-axis is the line of reflection. So, plot each vertex of G H J the same distance from the y-axis as its corresponding vertex on GHJ. Point G is 4 units to the right of the y-axis, so point G is 4 units to the left of the y-axis. Point H is 3 units to the right of the y-axis, so point H is 3 units to the left of the y-axis. Point J is 1 unit to the right of the y-axis, so point J is 1 unit to the left of the y- axis. The algebraic representation (x, y) ( x, y) explains the effect of the reflection. So, the coordinates are G ( 4, 2), H ( 3, 4), and J ( 1, 1). Analyze Relationships The coordinates of point M are (3, 3). The coordinates of its image after a reflection are M (3, 3). Describe the reflection as over the x-axis or y-axis. In the reflection M(, ) M (3, ), the x-coordinate stays the same and the y-coordinate changes from 3 to 3. This indicates a reflection over the x-axis.
3 Lesson 8-3 Rotations Page 581 Triangle RST represents the placement of Tyra s tricycle in the driveway and has vertices R( 3, 4), S( 3, 1), and T(0, 1). Graph the figure and its rotated image after a clockwise rotation of 180º about the origin. Then determine the coordinates of the vertices for triangle R S T. Graph RST on a coordinate plane. After graphing, sketch segment TO connecting point T to the origin. Sketch another segment, TO, ' so that the angle between point T, O, and T measures 180 clockwise and the segment is the same length as TO. Repeat for points R and S. Then connect the vertices to form R S T. The coordinates of the vertices for triangle R S T are R (3, 4), S (3, 1), and T (0, 1).
4 The right isosceles triangle PQR has vertices P(3, 3), Q(3, 1), and R(x, y) and is rotated 90 counterclockwise about the origin. Determine the coordinates of vertex R. Then graph the triangle and its image. Graph the two known vertices P(3, 3) and Q(3, 1), and connect the points. Since PQR is a right isosceles triangle, graph a point R so that the vertices form a right isosceles triangle. Sample coordinates: R(1, 1). Next, sketch segment RO connecting point R to the origin. Sketch another segment, RO, ' so that the angle between point R, O, and R measures 90 counterclockwise and the segment is the same length as RO.
5 Repeat for points P and Q. Then connect the vertices to form P Q R.
6 Lesson 8-4 Congruence and Transformations Page 593 Determine if the two figures are congruent by using transformations. Explain your reasoning. Rotate parallelogram RSTU 90º clockwise. Label the vertices of the image R, S, T, and U. Compare it to parallelogram WXYZ. Even if the rotated figure is translated right, it will not have the same size as parallelogram WXYZ. So, the two figures are not congruent since no sequence of transformations maps parallelogram RSTU onto parallelogram WXYZ.
7 Nilda purchased some custom printed stationery with her initials. What transformations could be used if the letter Z is the preimage and the letter N is the image in the design shown? Are the two figures congruent? Explain. Start with the preimage. Rotate the letter Z 90º counterclockwise about point A. A Translate the new image to the left and down. So, a 90º counterclockwise rotation followed by a translation could be used to create the design. The letters are congruent because an image produced by a rotation and a translation have the same size and shape.
8 Lesson 8-5 Congruence Page 605 Write congruent statements comparing the corresponding parts in the congruent figures below. Use the matching arcs and tick marks to identify the corresponding parts. Corresponding angles: N S, M T, O V Corresponding sides: ON VS, NM ST, MO TV Parallelograms UVWX and HJIK are congruent. Write congruence statements comparing the corresponding parts. Then determine which transformation(s) map parallelogram UVWX onto parallelogram HJIK. First, analyze the figures to determine which angles and sides of the figures correspond. Corresponding angles: Corresponding sides: Next, determine any changes in the orientation of the parallelograms. Since the orientation is reversed, at least one of the transformations is a reflection. Sample answer: If you reflect paralellogram UVWX over the x-axis, then translate it 4 units to the right, it coincides with parallelogram HJIK. So, the transformations that map parallelogram UVWX onto parallelogram HJIK consist of a reflection over the x-axis followed by a translation of 4 units to the right.
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