Composing Figures

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Exploring Translations in the coordinate plane.

(A). Here is a triangle. Reflect triangle ABC over line AB . Label the image of C as C’.

(B). Rotate triangle ABC’ around A so that C’ matches up with B.

What can you say about the measures of angles B and C ?

(C). Here is isosceles triangle ONE. Its sides ON and OE have equal lengths. Angle O is 30 degrees. The length of ON is 5 units. Reflect triangle ONE across segment ON. Label the new vertex M.

(1). What is the measure of angle MON ?

(2). What is the measure of angle MOE ?

(D). Reflect triangle MON across segment OM. Label the point that corresponds to N as T.

(1). How long is  ? How do you know?

(2). If you continue to reflect each new triangle this way to make a pattern as below, what will the pattern look like?

(3). What is the measure of angle TOE ?

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Exploring Translations in the coordinate plane.

(A). Here is triangle ABC. Draw midpoint M of side AC. Rotate triangle ABC 180 degrees using center M to form triangle CDA. Draw and label this triangle.

(1). What kind of quadrilateral is ABCD? Explain how you know.

In the activity, we made a parallelogram by taking a triangle and its image under a 180-degree rotation around the midpoint of a side. This picture helps you justify a well-known formula for the area of a triangle.

(2). What is the formula and how does the figure help justify it?

(B). The picture shows 3 triangles. Triangle 2 and Triangle 3 are images of Triangle 1 under rigid transformations.

(1). Describe a rigid transformation that takes Triangle 1 to Triangle 2.

(2). What points in Triangle 2 correspond to points A, B, and C in the original triangle?

(3). Describe a rigid transformation that takes Triangle 1 to Triangle 3.

(4). What points in Triangle 3 correspond to points A, B, and C ?

(C). Find congruent line segments and angles.

(1). From the diagram above, find two pairs of line segments that have the same measure.

(2). Explain how you know the lines have the same measure.

(3). From the diagram above, find two pairs of angles that have the same measure.

(4). Explain how you know the angles have the same measure.

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Exploring Rotations of a Figure about a Point on the Figure.

Here below is triangle XYZ:

(1). Rotate triangle XYZ 90 degrees clockwise around Z.

(2). Rotate triangle XYZ 180 degrees clockwise around Z.

(3). Rotate triangle XYZ 270 degrees clockwise around Z.

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Here is the design of an older version of the flag of Great Britain. There is a rigid transformation that takes Triangle 1 to Triangle 2, another that takes Triangle 1 to Triangle 3, and another that takes Triangle 1 to Triangle 4.

(1). Measure the lengths of the sides in Triangles 1 and 2. What do you notice?

(2). What are the side lengths of Triangle 3?

(3). Explain how you know.

(4). Do all eight triangles in the flag have the same area? Explain how you know.

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Study the lines in the figure above.

(2). Explain how you know.

(3). Explain how to translate, rotate or reflect line to obtain line .

(4). Explain how to translate, rotate or reflect line to obtain line .

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Point A has coordinates (3, 4).

After a translation of 4 units to the left, a reflection across the -axis, and a translation of 2 units down, what will be the coordinates of the image?

, , and in the original triangle?