geometry - mesa public schools · or a composition of rigid motions that maps one of the ... then...

Geometry

4.4 Congruence and Transformations

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


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?



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

△DEF ≅ △ABC; △DEF is a 90° rotation of △ABC.


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


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


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



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 = x (𝒂, 𝒃) (𝒃, 𝒂)

Rotate 90 CW

Rotate 90 CCW

Rotate 180


Mapping Formulas

Mapping Formula





Rotate 90 CW (𝒂, 𝒃) (𝒃, −𝒂)

Rotate 90 CCW

Rotate 180


Mapping Formulas

Mapping Formula






Rotate 90 CCW (𝒂, 𝒃) (−𝒃, 𝒂)

Rotate 180


Mapping Formulas

Mapping Formula






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

A

B

1.Reflect AB in the y-axis.

2.Reflect A’B’ in the x-axis.

A’

B’

A’’

B’’


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’’


The order doesn’t matter.

A

B

A’

B’

A’’

B’’

A’

B’

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.

Reflect R’S’ in x-axis.

R

S

R’

S’

R’’

S’’


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.


▱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 5 units down.

Reflect ▱ABCD in the y-axis.

Example 5c


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.


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).


Reflections and Translations

Begin with ABC.

A C

B



Draw line of reflection m.

A C

B

m



Reflect the figure in the line.

A C

B

A’C’

B’

m



Draw line of reflection n parallel to m.

A C

B

A’C’

B’

m n



Reflect A’B’C’in line n.

A C

B

A’C’

B’

A’’ C’’

B’’

m n



A C

B

A’C’

B’

A’’ C’’

B’’

m n

A’’B’’C’’ has the same orientation as ABC.



A C

B

A’C’

B’

A’’ C’’

B’’

m n

Reflecting ABC twice is equal to a translation.


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.



A C

B

A’C’

B’

A’’ C’’

B’’

m n

d

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.

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.


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.



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



Furthermore, the amount of the rotation is twice the measure of the angle between lines k and m.

P

m

k

45

90



The amount of the rotation is twice the measure of the angle between lines k and m.

P

m

k

x

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° CCW 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

geometry - mesa public schools · or a composition of rigid motions that maps one of the ... then...

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