Back to Exercise: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations

Exercises: Congruence Through Sequences of Rigid Motions

Work through each section in order. When you describe a sequence of rigid motions, name each transformation in order and give its parameters: direction and distance for a translation, the line of reflection for a reflection, and the center and angle (with direction) for a rotation. Two figures are congruent only when a sequence of rigid motions maps one exactly onto the other.

Grade 8·21 problems·~35 min·Common Core Math - Grade 8·container·8-g-a-2

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Fluency Practice

For each pair of congruent figures, describe a sequence of rigid motions that maps the first onto the second. Name each transformation in order, with its parameters.

$A coordinate grid showing triangle ABC near the bottom and a congruent dashed triangle DEF directly above it.$

Triangle $ABC$ has vertices $A(1, 1)$ , $B(4, 1)$ , $C(2, 3)$ . Triangle $DEF$ has vertices $D(1, 6)$ , $E(4, 6)$ , $F(2, 8)$ . Describe a single rigid motion that maps $ABC$ onto $DEF$ , and explain why this proves $ABC \cong DEF$ .

$A coordinate grid showing solid triangle ABC in the first quadrant and a congruent dashed triangle DEF turned around in the third quadrant.$

Triangle $ABC$ has vertices $A(1, 1)$ , $B(4, 1)$ , $C(1, 3)$ . Triangle $DEF$ has vertices $D(-1, -1)$ , $E(-4, -1)$ , $F(-1, -3)$ . A single motion is not enough. Describe a sequence of two rigid motions that maps $ABC$ onto $DEF$ , naming each step in order.

$A coordinate grid showing triangle ABC on the right of the y-axis and its mirror-image dashed triangle DEF on the left.$

Triangle $ABC$ has vertices $A(1, 1)$ , $B(4, 1)$ , $C(2, 4)$ . Triangle $DEF$ has vertices $D(-1, 1)$ , $E(-4, 1)$ , $F(-2, 4)$ . Describe a single rigid motion that maps $ABC$ onto $DEF$ .

$A coordinate grid showing triangle GHI pointing right and a congruent dashed triangle JKL pointing up, sharing the origin.$

Triangle $GHI$ has vertices $G(0, 0)$ , $H(3, 0)$ , $I(1, 2)$ . Triangle $JKL$ has vertices $J(0, 0)$ , $K(0, 3)$ , $L(-2, 1)$ . They share the vertex $G = J$ . Which single rigid motion maps $GHI$ onto $JKL$ ?

A translation $3$ units up, since one vertex already shares the origin.

A reflection across the x-axis.

A rotation of $90\degree$ clockwise about the origin.

A rotation of $90\degree$ counterclockwise about the origin.

Point $P(2, 5)$ is rotated $90\degree$ clockwise about the origin using the rule $(x, y) \rightarrow (y, -x)$ . What are the coordinates of the image point $P'$ ? Write your answer in the form $(a, b)$ .

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Exercises: Congruence Through Sequences of Rigid Motions

Recall / Warm-Up

Fluency Practice