Geometry - Mesa Public Schools · or a composition of rigid motions that maps one of the ... then...
Transcript of Geometry - Mesa Public Schools · or a composition of rigid motions that maps one of the ... then...
4.4 Warm Up Day 1
11/23/2015 4.4 Conqruence and Transformations
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
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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 you make about a figure reflected in two lines?
11/23/2015 4.4 Conqruence and Transformations
11/23/2015 4.4 Conqruence and Transformations
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.
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Identifying Congruent Figures
Same size and shape
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different sizes or shapes
Example 1
Identify any congruent figures in the coordinate plane. Explain.
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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
△DEF ≅ △ABC; △DEF is a 90° rotation of △ABC.
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Identify any congruent figures in the coordinate plane. Explain.
△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 Formula
Translation
Reflect in y-axis
Reflect in x-axis
Reflect in y = x
Rotate 90 CW
Rotate 90 CCW
Rotate 180
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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
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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
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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
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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
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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
11/23/2015 4.4 Conqruence and Transformations
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
11/23/2015 4.4 Conqruence and Transformations
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)
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11/23/2015 4.4 Conqruence and Transformations
Compositions
A composition is a transformation that consists of two or more transformations performed one after the other.
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Example 3
A
B
1.Reflect AB in the y-axis.
2.Reflect A’B’ in the x-axis.
A’
B’
A’’
B’’
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Try it in a different order.
A
B
1.Reflect AB in the x-axis.
2.Reflect A’B’ in the y-axis.
A’
B’
A’’
B’’
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The order doesn’t matter.
A
B
A’
B’
A’’
B’’
A’
B’
This composition is commutative.
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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.
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Are all compositions commutative?
Rotate RS 90 CW.
Reflect R’S’ in x-axis.
R
S
R’
S’
R’’
S’’
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Reverse the order.
Reflect RS in the x-axis.
Rotate R’S’ 90 CW.
R
S
R’
S’
R’’
S’’
All compositions are NOT commutative. Order matters!
Example 5a
Describe a congruence transformation that maps ▱ABCD to ▱EFGH.
11/23/2015 4.4 Conqruence and Transformations
▱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.
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Translate it 5 units down.
Reflect ▱ABCD in the y-axis.
Example 5c
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Reflect ▱ABCD in the x-axis.
Then translate it 5 units left.
Why doesn’t the following work to transform ▱ABCD to ▱EFGH?
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.
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Reflection in the x-axis followed by a translation 5 units right. Or…
Translation 5 units right followed by areflection 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).
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Reflections and Translations
Draw line of reflection m.
A C
B
m
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Reflections and Translations
Reflect the figure in the line.
A C
B
A’C’
B’
m
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Reflections and Translations
Draw line of reflection n parallel to m.
A C
B
A’C’
B’
m n
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Reflections and Translations
Reflect A’B’C’in line n.
A C
B
A’C’
B’
A’’ C’’
B’’
m n
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Reflections and Translations
A C
B
A’C’
B’
A’’ C’’
B’’
m n
A’’B’’C’’ has the same orientation as ABC.
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Reflections and Translations
A C
B
A’C’
B’
A’’ C’’
B’’
m n
Reflecting ABC twice is equal to a translation.
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Theorem
If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is a translation.
If P’’ is the image of P, then PP’’ = 2d, where d is the distance between lines k and m.
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Reflections and Translations
A C
B
A’C’
B’
A’’ C’’
B’’
m n
d
2d
Example 6
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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.
b. Does AC = BD? Explain.
c. What is the length of GG″?
a. GH ≅ G ′H′ ≅ G ″H″ . HB ≅ H ′B . GA ≅ G ′A
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.
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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.
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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.
P
m
k
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Compound Reflections
Furthermore, the amount of the rotation is twice the measure of the angle between lines k and m.
P
m
k
45
90
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Compound Reflections
The amount of the rotation is twice the measure of the angle between lines k and m.
P
m
k
x
2x
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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° CCW rotation about point P.
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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.