Geometry. 4.4 Congruence and Transformations
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1 Geometry 4.4 Congruence and Transformations
2 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)
3 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 4.4 Essential Question What conjectures can I make about a figure reflected in two lines?
5 Goals Identify congruent figures. Describe congruence transformations. Use theorems about congruence transformations.
6 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.
7 Identifying Congruent Figures Same size and shape different sizes or shapes
8 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.
9 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.
10 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
11 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
12 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
13 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
14 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
15 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
16 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
17 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)
18 Compositions A composition is a transformation that consists of two or more transformations performed one after the other.
19 Example 3 1.Reflect AB in the y-axis. A A 2.Reflect A B in the x-axis. B B B A
20 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
21 The order doesn t matter. A A B B B B A A This composition is commutative.
22 Commutative Property a + b = b + a = ab = ba 4 25 = 25 4 Reflect in y, reflect in x is equivalent to reflect in x, reflect in y.
23 Are all compositions commutative? Rotate RS 90 CW. R R Reflect R S in x-axis. S S S R
24 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!
25 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.
26 Example 5b 8b. Describe another congruence transformation that maps ABCD to EFGH. Translate it 4 units down. Reflect ABCD in the y-axis.
27 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.
28 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).
29 Reflections and Translations Begin with ABC. B A C
30 Reflections and Translations Draw line of reflection m. m B A C
31 Reflections and Translations Reflect the figure in the line. m B B A C C A
32 Reflections and Translations m n Draw line of reflection n parallel to m. B B A C C A
33 Reflections and Translations m n Reflect A B C in line n. B B B A C C A A C
34 Reflections and Translations m n A B C has the same orientation as ABC. B B B A C C A A C
35 Reflections and Translations m n Reflecting ABC twice is equal to a translation. B B B A C C A A C
36 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.
37 Reflections and Translations m n d B B B A C C A A C 2d
38 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
39 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.
40 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
41 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
42 Compound Reflections The amount of the rotation is twice the measure of the angle between lines k and m. k m x P 2x
43 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
44 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.
45 Assignment
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