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**Exploring Line Translations in the coordinate plane.**

**(A).** In the applet below, describe the transformation that takes line to Then plot and label A’ and B’, the images of A and B.

**(B).** In the applet below, describe the transformation that takes line to Then plot and label A’ and B’, the images of A and B.

**(C).** Translate lines a and b 3 units up and 2 units to the right.

(1). What do you notice about the changes that occur to lines a and b after the translation?

(2). What is the same in the original and the image?

**(D).** Rotate lines a and b counterclockwise 180 degrees using K as the center of rotation.

(1). What do you notice about the changes that occur to lines a and b after the rotation?

(2). What is the same in the original and the image?

**(E).** Reflect lines and b across line h.

(1). What do you notice about the changes that occur to lines a and b after the reflection?

(2). What is the same in the original and the image?

(3). As you performed each transformation: What is the image of the two parallel lines under a rigid transformation?

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**Exploring Line Rotations in the coordinate plane.**

**(A). **The diagram shows a line with points labeled A, C, D, and B. On the diagram, draw the image of the line and points A, C, and B after the line has been rotated 180 degrees around point D. Label the images of the points A’, B’, and C’.

(1). What is the order of all seven points? Explain or show your reasoning.

**(B). **The diagram shows a line with points A and C on the line and a segment AD where D is not on the line. Rotate the figure 180 degrees about point C. Label the image of A as A’ and the image of D as D’.

What do you know about the relationship between angle CAD and angle C’A’D’ ? Explain or show your reasoning.

**(C). **The diagram shows two lines l and m that intersect at a point O with point A on l and point D on m. Rotate the figure 180 degrees around O. Label the image of A as A’ and the image of D as D’.

What do you know about the relationship between the angles in the figure? Explain or show your reasoning.

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**Applications of Transformations in the coordinate plane.**

**(A).** When you rotate two parallel lines, sometimes the two original lines intersect their images and form a quadrilateral.

What is the most specific thing you can say about this quadrilateral? Can it be a square? A rhombus? A rectangle that isn’t a square? Explain your reasoning.

**(B).** Use the diagram to find the measure of the angle indicated. Explain your reasoning.

(1). m∠**ABC** = ?

(2). m∠**EBD** = ?

(3). m∠**ABE** = ?

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In the picture, triangle A’B’C’ is an image of triangle ABC after a rotation. The center of rotation is D.

(1). What is the length of side ? Explain how you know.

(2). What is the measure of angle ? Explain how you know.

(3). What is the measure of angle ? Explain how you know.

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Points P and Q are plotted on a line. Move point R so that a 180-degree rotation with center R sends P to Q and Q to P.

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