Back to Exercise: Use rigid motions for congruence

Exercises: Use Rigid Motions to Define and Determine Congruence

Work through each section in order. For problems that ask you to find a rigid motion,
verify your answer by checking every vertex.

Grade 9·20 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-co-b-6

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Warm-Up: Review What You Know

These problems review skills you already know from earlier in the unit.

Which of the following is the formal definition of a rigid motion?

A transformation that preserves all distances and angle measures between points

A transformation that moves a figure to the same position and orientation

A transformation that preserves shape but may change size

A transformation that includes translations, rotations, reflections, and dilations

Triangle $ABC$ has vertices $A(1, 2)$ , $B(4, 2)$ , and $C(1, 5)$ .
After a translation by vector $\langle 3, -4 \rangle$ , what are the coordinates
of the image vertex $A'$ ?

$(-2, 6)$

$(4, -2)$

$(3, -4)$

$(1, -2)$

Which property do all rigid motions (translations, reflections, rotations) share?

They always preserve the orientation (handedness) of a figure

They always move the figure to a new position

They preserve all distances between corresponding points

They require a center point or line to be defined

Fluency Practice

Apply rigid motions and the definition of congruence to answer each problem.

According to the rigid-motion definition, two figures are congruent if and only if:

They have the same perimeter and area

They are in the same position and have the same orientation

There exists a sequence of rigid motions mapping one figure exactly onto the other

There exists a sequence of rigid motions or dilations mapping one figure
onto the other

$Coordinate plane showing triangle PQR with vertices P(2,1), Q(5,1), R(2,4) above the x-axis, with the x-axis marked as the line of reflection.$

Triangle $PQR$ has vertices $P(2, 1)$ , $Q(5, 1)$ , and $R(2, 4)$ .
It is reflected over the $x$ -axis. In which quadrant does vertex $R'$ land,
and does the reflection reverse the orientation of the triangle?

Quadrant III; yes, the reflection reverses orientation

Quadrant IV; yes, the reflection reverses orientation

Quadrant IV; no, the reflection preserves orientation

Quadrant I; no, the reflection preserves orientation

$Coordinate plane showing triangle ABC with vertices A(2,1), B(4,1), C(2,4) in Quadrant I and a counterclockwise rotation arrow at the origin.$

Triangle $ABC$ has vertices $A(2, 1)$ , $B(4, 1)$ , and $C(2, 4)$ .
It is rotated $90^\circ$ counterclockwise about the origin. Which quadrant
contains the image vertex $A'$ ?

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

Triangle $ABC$ has vertices $A(0, 0)$ , $B(3, 0)$ , $C(0, 4)$ .
A student claims that to verify a rigid motion maps $ABC$ to a target triangle,
it is sufficient to check that the three vertex images match the three target vertices.
Is this claim correct?

No — every point on every side must be checked individually

No — only the midpoints of each side need to be checked in addition
to the vertices

Yes — rigid motions map line segments to line segments, so if all vertices
match, all points on the sides automatically match

Yes — but only for triangles; for quadrilaterals, you must check diagonals
as well

Triangle $DEF$ has vertices $D(-1, -1)$ , $E(-4, -1)$ , and $F(-2, -4)$ .
Triangle $GHI$ has vertices $G(1, 1)$ , $H(4, 1)$ , and $I(2, 4)$ .
Which single rigid motion maps $\triangle DEF$ onto $\triangle GHI$ ?

A translation by $\langle 2, 2 \rangle$

A $180^\circ$ rotation about the origin

A reflection over the $y$ -axis

A reflection over the line $y = x$

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