Exercises: Use rigid motions for congruence - Common Core Math - HS Geometry

A

Warm-Up: Review What You Know

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

1.

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

A.

A transformation that preserves all distances and angle measures between points

B.

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

C.

A transformation that preserves shape but may change size

D.

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

2.

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 $A'$ ?

A.

$(-2, 6)$

B.

$(4, -2)$

C.

$(3, -4)$

D.

$(1, -2)$

3.

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

A.

They always preserve the orientation (handedness) of a figure

B.

They always move the figure to a new position

C.

They preserve all distances between corresponding points

D.

They require a center point or line to be defined

B

Fluency Practice

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

C

Mixed Practice

These problems test the same skills in different ways.

D

Word Problems

Read each scenario. Use the rigid-motion definition of congruence to answer the questions.

1.

A design program generates two triangles on a coordinate grid.
Triangle $ABC$ has vertices $A(1, 1)$ , $B(4, 1)$ , and $C(1, 5)$ .
Triangle $DEF$ has vertices $D(-1, -1)$ , $E(-4, -1)$ , and $F(-1, -5)$ .
A programmer wants to confirm the two triangles are congruent by finding a
rigid motion.

1.

As a preliminary check, which rigid motion should the programmer test first,
based on the positions of the two triangles?

A.

A translation, because the triangles are in different locations

B.

A reflection over the $y$ -axis

C.

A $180^\circ$ rotation about the origin, because $\triangle DEF$ appears
to be the image of $\triangle ABC$ rotated half a turn

D.

A dilation with scale factor $-1$

2.

Verify the rigid motion from part (a) by showing the image of each vertex
of $\triangle ABC$ under a $180^\circ$ rotation about the origin, and
confirm it matches the corresponding vertex of $\triangle DEF$ .

3.

The programmer finds another sequence of rigid motions that also maps
$\triangle ABC$ onto $\triangle DEF$ (a reflection followed by a rotation).
What does this imply about the congruence of the triangles?

A.

It creates a contradiction — two rigid motions cannot both map the same
pair of figures, so there must be an error

B.

It implies the triangles might not actually be congruent, since the
rigid motion from part (a) might be wrong

C.

It is consistent with the definition — congruence requires at least one
rigid motion, not a unique one, so multiple solutions are valid

D.

It means the triangles are "more congruent" than if only one rigid
motion existed

E

Find the Mistake

Each problem shows student reasoning that contains an error.
Identify the mistake and choose the best explanation.

1.

Alex looked at two triangles:
Triangle $PQR$ with vertices $P(1, 2)$ , $Q(3, 2)$ , $R(1, 5)$ , and
Triangle $P'Q'R'$ with vertices $P'(4, 2)$ , $Q'(6, 2)$ , $R'(4, 5)$ .

Alex said: "These triangles are NOT congruent because they are in different
positions — a congruent figure must be in the same place."

Which statement best identifies Alex's error?

A.

Alex is correct — congruent figures must occupy the same position in
the plane

B.

Alex confused congruence with identity. Congruent figures may be in
different positions; what matters is whether a rigid motion maps one
onto the other

C.

Alex is correct that they are not congruent, but for the wrong reason —
the triangles are not congruent because they have different orientations

D.

Alex should have checked whether a dilation maps one triangle onto the other,
which would establish congruence

2.

Jamie was given two triangles on a coordinate grid and wrote:

"Triangle $ABC$ and Triangle $DEF$ look the same size to me when I compare them
visually, so they must be congruent. I don't need to find a rigid motion or measure
anything — my eyes tell me they are the same shape and size."

What is the fundamental flaw in Jamie's reasoning?

A.

Jamie's reasoning is valid — visual comparison on a coordinate grid is
a reliable test for congruence

B.

Jamie should measure only the areas of the two triangles to confirm congruence

C.

Visual appearance is not a formal argument. Congruence must be established
by exhibiting a rigid motion or verifying all corresponding measurements
are exactly equal — diagrams can be misleading

D.

Jamie should check that the triangles have the same perimeter, which is the
correct formal test for congruence

F

Challenge Problems

These problems require multi-step reasoning. Explain your thinking clearly.

1.

Figure $X$ is congruent to figure $Y$ via rigid motion $f$ (a reflection over
the $y$ -axis). Figure $Y$ is congruent to figure $Z$ via rigid motion $g$
(a translation by $\langle 3, 0 \rangle$ ). Describe a single sequence of
rigid motions that maps $X$ directly onto $Z$ , and explain why this sequence
proves $X \cong Z$ without needing to compare $X$ and $Z$ directly.

2.