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9.5 Applying Compositions of Transformations. Composition of Transformation- 2 or more transformations are combined to form a single transformation The composition of 2 (or more) isometries is an isometry .

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9 5 applying compositions of transformations
9.5 Applying Compositions of Transformations
  • Composition of Transformation- 2 or more transformations are combined to form a single transformation
  • The composition of 2 (or more) isometries is an isometry.
slide2
Glide Reflection:1. Translate P P΄2. Reflect P΄ P΄΄ in a line k parallel to the direction of the translation

P΄΄

k

P

slide3

Begin by graphing ABC. Then graph A′B′C′ after a translation 12 units left. Finally, graph A′′B′′C′′ after a reflection in the x-axis.

Find the image of a glide reflection

The vertices of ABCare A(3, 2), B(6, 3), and C(7, 1). Find the image of ABCafter the glide reflection.

Translation:(x, y) →

(x –12, y)

Reflection: in the x-axis

SOLUTION

slide4

The endpoints of RSare R(1, –3) and S(2, –6). Graph the image of RSafter the composition.

STEP 1

Graph RS

STEP 2

Reflect RSin the y-axis. R′S′ has endpoints R′(–1, –3) and S′(–2, –6).

Find the image of a composition

Reflection: in the y-axis

Rotation: 90° about the origin

SOLUTION

slide5

Find the image of a composition

o

Rotate R′S′ 90 about the origin. R′′S′′ has endpoints R′′(3, –1) andS′′(6, –2).

STEP 3

reflections in parallel lines theorem 9 5
Reflections in Parallel Lines Theorem (9.5)
  • If lines k and m are parallel, then a reflection in k followed by a reflection in line m is the same as a translation
slide7

In the diagram, a reflection in line kmaps GH to G′H′. A reflection in line m maps G′H′to G′′H′′. Also, HB = 9 andDH′′ = 4.

What is the length of GG′′ ?

By the properties of reflections, H′B = 9 and H′D = 4. Theorem 9.5 implies that GG′′ = HH′′ = 2 BD, so the length of GG′′ is 2(9 + 4), or 26 units.

Use Theorem 9.5

SOLUTION

slide8

ANSWER

The preimage is reflected in line k , then in line m. Describe a single transformation that maps the blue figure to the green figure.

Translation

More Examples

The distance between line k and line mis 1.6 centimeters.

slide9

What is the distance between Pand P′′? If you draw PP′ , what is its relationship with line k? Explain.

ANSWER

3.2 cm;

They are perpendicular.

The distance between line k and line mis 1.6 centimeters

reflections in intersecting lines theorem 9 6
Reflections in Intersecting Lines Theorem 9.6
  • 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.
  • The angle of rotation is 2x˚, where x˚is the

measure of the acute or right angle formed

by k and m

m

P

P

k

  • P
slide11

Use Theorem9.6

In the diagram, the figure is reflected in line k.The image is then reflected in line m. Describe a single transformation that maps Fto F′′.

slide12

Use Theorem9.6

SOLUTION

The measure of the acute angle formed between lines kand mis 70° . So, by Theorem 9.6, a single transformation that maps Fto F′′ is a 140° rotation about point P.

You can check that this is correct by tracing lines kand mand point F, then rotating the point 140° .

slide13

In the diagram at the right, the preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure onto the green figure.

ANSWER

A rotation of 160°about point P

Use Theorem 9.6