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Back to Exercise: Lines are taken to lines, and line segments to line segments of the same length

Exercises: Rigid Motions Preserve Lines and Lengths

Work through each section in order. For length calculations, use the distance formula d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. Show your work where indicated.

Grade 8·20 problems·~30 min·Common Core Math - Grade 8·standard·8-g-a-1a
Work through problems with immediate feedback
A

Recall / Warm-Up

These problems review skills you already know: plotting points, finding lengths, and applying transformation rules.

A coordinate grid with a segment from point A at (1, 1) to point B at (4, 5).
1

A segment has endpoints A(1,1)A(1, 1) and B(4,5)B(4, 5). Using the distance formula, what is the length of segment ABAB?

2

Which list of transformations contains only rigid motions (transformations that keep a figure the same size and shape)?

3

Apply the translation rule (x,y)→(x+2,y−3)(x, y) \rightarrow (x + 2, y - 3) to the point (5,4)(5, 4). What is the image point?

B

Fluency Practice

Apply each transformation and verify the line- and length-preservation properties. Show your work.

A coordinate grid showing five original points on a rising line with a dashed arrow indicating the translation direction; the image points are not shown.
1

The line through the points (−2,−1),(−1,0),(0,1),(1,2),(2,3)(-2, -1), (-1, 0), (0, 1), (1, 2), (2, 3) is translated by the rule (x,y)→(x+4,y−2)(x, y) \rightarrow (x + 4, y - 2). After translating all five points, what is true about the images?

A coordinate grid showing segment AB and its translated image segment of the same length.
2

Segment ABAB has endpoints A(1,2)A(1, 2) and B(4,6)B(4, 6), with length 55. Apply the translation (x,y)→(x+3,y−1)(x, y) \rightarrow (x + 3, y - 1) to get A′A' and B′B'. What is the length of the image segment A′B′A'B'?

3

Segment PQPQ has endpoints P(2,1)P(2, 1) and Q(2,6)Q(2, 6). It is rotated 180°180\degree about the origin to give P′(−2,−1)P'(-2, -1) and Q′(−2,−6)Q'(-2, -6). What is the length of the image segment P′Q′P'Q'?

A coordinate grid showing segment CD and its reflection across the y-axis, both the same length.
4

Segment CDCD has endpoints C(−3,2)C(-3, 2) and D(1,−1)D(1, -1). It is reflected across the y-axis to give C′(3,2)C'(3, 2) and D′(−1,−1)D'(-1, -1). What is the length of the image segment C′D′C'D'?

5

Segment EFEF has endpoints E(0,0)E(0, 0) and F(2,3)F(2, 3). Its length is 13\sqrt{13}. After a translation, the image endpoints are E′(5,1)E'(5, 1) and F′(7,4)F'(7, 4). What is the length of E′F′E'F'? Give your answer as an exact value.

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