Geometry 4.4 Congruence and Transformations
4.4 Warm Up Day 1 Plot and connect the points in a coordinate plane to make a polygon. Name the polygon. 1. A(-3, 2), B(-2, 1), C(3, 3) 2. E(1, 2), F(3, 1), G(-1, -3), H(-3, -2) 3. J(3, 3), K(3, -3), L(-3, -3), M(-3, 3) 4. P(2, -2), Q(4, -2), R(5, -4), S(2, -4)
4.4 Warm Up Day 2 Graph ABC. Graph each transformation to ABC and state the coordinates of the images. A(-5, 3), B(-4, -1), C(-2, 1) 1. (x, y) (x+6, y-3) A' ( ), B( ), C'( ) 2. Rotate ABC 90 CW A' ( ), B( ), C'( ) 3. Reflect ABC onto the x-axis A^' ( ), B( ), C'( )
4.4 Essential Question What conjectures can I make about a figure reflected in two lines?
Goals Identify congruent figures. Describe congruence transformations. Use theorems about congruence transformations.
Identifying Congruent Figures Two figures are congruent if and only if there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other. Congruent figures have the same size and shape.
Identifying Congruent Figures Same size and shape different sizes or shapes
Example 1 Identify any congruent figures in the coordinate plane. Explain. Square NPQR square ABCD Square NPQR is a translation of square ABCD 2 units left and 6 units down. EFG KLM KLM is a reflection of EFG in the x-axis. HIJ STU STU is a 180 rotation of HIJ.
Your Turn Identify any congruent figures in the coordinate plane. Explain. DEF ABC; DEF is a 90 rotation of ABC. KLM STU; KLM is a reflection of STU in the y-axis. GHIJ NPQR; GHIJ is a translation 6 units up of NPQR.
Mapping Formulas Mapping Translation Reflect in y-axis Reflect in x-axis Reflect in y = x Rotate 90 CW Rotate 90 CCW Rotate 180 Formula
Mapping Formulas Mapping Formula Translation (x, y) (x + a, y + b) Reflect in y-axis Reflect in x-axis Reflect in y = x Rotate 90 CW Rotate 90 CCW Rotate 180
Mapping Formulas Mapping Formula Translation (x, y) (x + a, y + b) Reflect in y-axis (a, b) ( a, b) Reflect in x-axis Reflect in y = x Rotate 90 CW Rotate 90 CCW Rotate 180
Mapping Formulas Mapping Formula Translation (x, y) (x + a, y + b) Reflect in y-axis (a, b) ( a, b) Reflect in x-axis (a, b) (a, b) Reflect in y = x Rotate 90 CW Rotate 90 CCW Rotate 180
Mapping Formulas Mapping Formula Translation (x, y) (x + a, y + b) Reflect in y-axis (a, b) ( a, b) Reflect in x-axis (a, b) (a, b) Reflect in y = x (a, b) (b, a) Rotate 90 CW Rotate 90 CCW Rotate 180
Mapping Formulas Mapping Formula Translation (x, y) (x + a, y + b) Reflect in y-axis (a, b) ( a, b) Reflect in x-axis (a, b) (a, b) Reflect in y = x (a, b) (b, a) Rotate 90 CW (a, b) (b, a) Rotate 90 CCW Rotate 180
Mapping Formulas Mapping Formula Translation (x, y) (x + a, y + b) Reflect in y-axis (a, b) ( a, b) Reflect in x-axis (a, b) (a, b) Reflect in y = x (a, b) (b, a) Rotate 90 CW (a, b) (b, a) Rotate 90 CCW (a, b) ( b, a) Rotate 180
Mapping Formulas Mapping Formula Translation (x, y) (x + a, y + b) Reflect in y-axis (a, b) ( a, b) Reflect in x-axis (a, b) (a, b) Reflect in y = x (a, b) (b, a) Rotate 90 CW (a, b) (b, a) Rotate 90 CCW (a, b) ( b, a) Rotate 180 (a, b) ( a, b)
Compositions A composition is a transformation that consists of two or more transformations performed one after the other.
Example 3 1.Reflect AB in the y-axis. A A 2.Reflect A B in the x-axis. B B B A
Try it in a different order. 1.Reflect AB in the x-axis. A 2.Reflect A B in the y-axis. B B B A A
The order doesn t matter. A A B B B B A A This composition is commutative.
Commutative Property a + b = b + a 25 + 5 = 5 + 25 ab = ba 4 25 = 25 4 Reflect in y, reflect in x is equivalent to reflect in x, reflect in y.
Are all compositions commutative? Rotate RS 90 CW. R R Reflect R S in x-axis. S S S R
Reverse the order. Reflect RS in the x-axis. R R Rotate R S 90 CW. S S S R All compositions are NOT commutative. Order matters!
Example 5a Describe a congruence transformation that maps ABCD to EFGH. ABCD and EFGH slant in opposite directions. If you reflect ABCD in the y-axis, then the image, A B C D, will have the same orientation as EFGH. Then you can map A B C D to EFGH using a translation of 4 units down.
Example 5b 8b. Describe another congruence transformation that maps ABCD to EFGH. Translate it 4 units down. Reflect ABCD in the y-axis.
Example 5c Why doesn t the following work to transform ABCD to EFGH? Reflect ABCD in the x-axis. Then translate it 5 units left. This transformation will match A with G, not with E. The other points don t match either.
Your turn a. Describe a congruence transformation that maps JKL to MNP. Reflection in the x-axis followed by a translation 5 units right. Or Translation 5 units right followed by a reflection in the x-axis b. Why doesn t a reflection in the y-axis followed by a reflection in the x-axis work? N would be at (3, -4), not at (2, -4).
Reflections and Translations Begin with ABC. B A C
Reflections and Translations Draw line of reflection m. m B A C
Reflections and Translations Reflect the figure in the line. m B B A C C A
Reflections and Translations m n Draw line of reflection n parallel to m. B B A C C A
Reflections and Translations m n Reflect A B C in line n. B B B A C C A A C
Reflections and Translations m n A B C has the same orientation as ABC. B B B A C C A A C
Reflections and Translations m n Reflecting ABC twice is equal to a translation. B B B A C C A A C
Theorem If lines m and n are parallel, then a reflection in line m followed by a reflection in line n is a translation. If P is the image of P, then PP = 2d, where d is the distance between lines m and n.
Reflections and Translations m n d B B B A C C A A C 2d
Example 6 In the diagram, a reflection in line k maps GH to G H. A reflection in line m maps G H to G H. Also, HB = 9 and DH = 4 a. Name any segments congruent to each segment: GH, HB, and GA. a. GH G H G H. HB H B. GA G A b. Does AC = BD? Explain. b. Yes, AC = BD because GG and HH are perpendicular to both k and m. So, BD and AC are opposite sides of a rectangle. c. What is the length of GG? GG = 2AC
Compound Reflections If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P.
Compound Reflections If lines k and m intersect at point P, then a reflection in k followed by a reflection in m is the same as a rotation about point P. k m P
Compound Reflections Furthermore, the amount of the rotation is twice the measure of the angle between lines k and m. k m 45 P 90
Compound Reflections The amount of the rotation is twice the measure of the angle between lines k and m. k m x P 2x
Example 7 In the diagram, the figure is reflected in line k. The image is then reflected in line m. Describe a single transformation that maps F to F The measure of the acute angle formed between lines k and m is 70. So, by the Reflections in Intersecting Lines Theorem, the angle of rotation is 2(70 ) = 140. A single transformation that maps F to F is a 140 rotation about point P
Summary Congruent figures have the same size and shape. Translation, reflection, and rotation are isometries (preimage and image are congruent figures) A composition consists of two or more transformations performed one after the other. The composition of rigid transformations gives congruent figures.
Assignment