Congruence Via Rigid Motions - Geometry Help

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Question 1

Two regular hexagons are positioned so that one can be mapped onto the other using exactly two rigid motions: a reflection followed by a rotation. If the center of the first hexagon is at $(0, 0)$ and the center of the second hexagon is at $(4, 3)$ , what can be concluded about the congruence of these figures?

The hexagons are congruent because any sequence of exactly two rigid motions always preserves congruence relationships
The hexagons are congruent because the combination of reflection and rotation forms a glide reflection preserving distances
The hexagons are not congruent because the centers are at different positions, violating the definition of congruent figures
The hexagons are congruent because rigid motions preserve shape and size regardless of the number of transformations used (correct answer)

Explanation: Two figures are congruent if one can be mapped onto the other by a sequence of rigid motions (translations, reflections, rotations). Since rigid motions preserve distances and angle measures, any sequence of rigid motions will result in congruent figures. The fact that two motions are needed and that the centers are in different positions doesn't affect congruence.

Question 2

Two congruent regular octagons are positioned in the coordinate plane such that one is centered at the origin and the other is centered at point $(5, 12)$ . Assuming both octagons have the same orientation, which single rigid motion maps the first octagon onto the second?

Translation by vector $(5, 12)$ , because moving from one center to another with same orientation requires only displacement (correct answer)
Rotation of $90°$ about the origin, because regular octagons have rotational symmetry allowing positional mapping
Reflection across the line $y = \frac{12}{5}x$ , because this line passes through both octagon centers
Glide reflection combining translation and reflection, because any change in position requires composite motion

Explanation: When two congruent figures have the same orientation and different positions, a translation by the displacement vector maps one onto the other. The vector from (0,0) to (5,12) is (5,12), so translation by this vector maps the first octagon onto the second.

Question 3

Triangle $PQR$ is reflected across line $\ell$ to produce triangle $P'Q'R'$ . Then triangle $P'Q'R'$ is reflected across line $m$ , where lines $\ell$ and $m$ intersect at point $O$ and form a $35°$ angle. The final image is triangle $P''Q''R''$ . What single rigid motion is equivalent to this composition of two reflections?

A rotation of $35°$ about point $O$ , because the angle between intersecting reflection lines equals the rotation angle
A rotation of $70°$ about point $O$ , because the composition of reflections across intersecting lines creates double the intersection angle (correct answer)
A translation by twice the distance between lines $\ell$ and $m$ , because intersecting reflections produce translation motions
A glide reflection combining $35°$ rotation with translation, because intersecting reflections create composite transformation motions

Explanation: The composition of two reflections across intersecting lines is equivalent to a rotation about the point of intersection. The angle of rotation is twice the acute angle between the lines. Since the lines form a 35° angle, the equivalent rotation is 2 × 35° = 70° about point O.

Question 4

Triangle $JKL$ is reflected across line $m$ to create triangle $J'K'L'$ , which is then rotated $60°$ clockwise about point $P$ to create triangle $J''K''L''$ . If triangle $MNO$ can be mapped onto triangle $J''K''L''$ using only a single reflection, what must be true about triangles $JKL$ and $MNO$ ?

They are congruent because the composition of reflection, rotation, and reflection always preserves congruence relationships completely
They are not congruent because an odd number of reflections changes orientation while preserving some distance relationships
They are congruent because any sequence of rigid motions connecting two figures through intermediates establishes congruence between endpoints (correct answer)
They are similar but not congruent because rotations followed by reflections create proportional rather than identical figures

Explanation: Since triangle JKL can be mapped to triangle J''K''L'' through rigid motions (reflection then rotation), and triangle MNO can be mapped to triangle J''K''L'' through a rigid motion (single reflection), then triangle JKL and triangle MNO are congruent. Congruence is transitive: if A ≅ C and B ≅ C, then A ≅ B.

Question 5

Parallelogram $WXYZ$ has vertices $W(-2, 1)$ , $X(1, 3)$ , $Y(4, 1)$ , and $Z(1, -1)$ . A student claims that this parallelogram is congruent to parallelogram $ABCD$ with vertices $A(3, -2)$ , $B(5, 1)$ , $C(8, -1)$ , and $D(6, -4)$ . To verify this claim using the definition of congruence in terms of rigid motions, which approach is most efficient?

Compare corresponding side lengths and angles, then attempt to construct a sequence of rigid motions if measurements match
Find a translation vector that maps one vertex pair, then check if remaining vertices align under the same translation
Calculate the area and perimeter of both parallelograms, then apply rigid motions only if these measurements are identical
Determine if there exists a sequence of rigid motions mapping one parallelogram onto the other by systematic transformation analysis (correct answer)

Explanation: The definition of congruence in terms of rigid motions states that two figures are congruent if and only if one can be mapped onto the other by a sequence of rigid motions. The most direct approach is to systematically attempt to find such a sequence. While option A is practical, option D directly applies the definition and is the most theoretically sound approach.

Question 6

Are the two figures congruent? Which rigid motion(s) justify your answer, based only on the diagram?

Diagram description: A coordinate plane is shown with two polygons.

The coordinate grid has uniform units with labeled axes.
Figure 1 is triangle $\triangle A B C$ with points $A(-4,1)$ , $B(-2,3)$ , and $C(-1,0)$ .
Figure 2 is triangle $\triangle D E F$ with points $D(4,1)$ , $E(2,3)$ , and $F(1,0)$ .
No side lengths or angle measures are marked beyond what can be inferred from the coordinates. Diagram not drawn to scale beyond the uniform grid.

Yes; reflecting Figure 1 across the $y$ -axis maps it onto Figure 2. (correct answer)
No; the triangles are on opposite sides of the $y$ -axis, so they cannot be congruent.
Yes; dilating Figure 1 by a factor of $-1$ maps it onto Figure 2.
Yes; translating Figure 1 up 2 units maps it onto Figure 2.

Explanation: The skill being assessed is congruence via rigid motions in geometry. Two figures are congruent if there exists a sequence of rigid motions—translations, rotations, and reflections—that maps one exactly onto the other, preserving all distances and angles. In this diagram, a reflection across the y-axis maps the coordinates of Triangle ABC exactly onto those of Triangle DEF. Rigid motions preserve distance and angle because they do not stretch, shrink, or bend the figure, maintaining the exact size and shape. Therefore, the figures are congruent, as the reflection aligns all points perfectly. A common misconception is that being on opposite sides prevents congruence, but reflections handle such symmetry. To apply this, imagine sliding, turning, or flipping one figure exactly onto the other to check for a perfect match.

All questions

Question 1

The hexagons are congruent because any sequence of exactly two rigid motions always preserves congruence relationships
The hexagons are congruent because the combination of reflection and rotation forms a glide reflection preserving distances
The hexagons are not congruent because the centers are at different positions, violating the definition of congruent figures
The hexagons are congruent because rigid motions preserve shape and size regardless of the number of transformations used (correct answer)

Question 2

Translation by vector $(5, 12)$ , because moving from one center to another with same orientation requires only displacement (correct answer)
Rotation of $90°$ about the origin, because regular octagons have rotational symmetry allowing positional mapping
Reflection across the line $y = \frac{12}{5}x$ , because this line passes through both octagon centers
Glide reflection combining translation and reflection, because any change in position requires composite motion

Question 3

A rotation of $35°$ about point $O$ , because the angle between intersecting reflection lines equals the rotation angle
A rotation of $70°$ about point $O$ , because the composition of reflections across intersecting lines creates double the intersection angle (correct answer)
A translation by twice the distance between lines $\ell$ and $m$ , because intersecting reflections produce translation motions
A glide reflection combining $35°$ rotation with translation, because intersecting reflections create composite transformation motions

Question 4

They are congruent because the composition of reflection, rotation, and reflection always preserves congruence relationships completely
They are not congruent because an odd number of reflections changes orientation while preserving some distance relationships
They are congruent because any sequence of rigid motions connecting two figures through intermediates establishes congruence between endpoints (correct answer)
They are similar but not congruent because rotations followed by reflections create proportional rather than identical figures

Question 5

Compare corresponding side lengths and angles, then attempt to construct a sequence of rigid motions if measurements match
Find a translation vector that maps one vertex pair, then check if remaining vertices align under the same translation
Calculate the area and perimeter of both parallelograms, then apply rigid motions only if these measurements are identical
Determine if there exists a sequence of rigid motions mapping one parallelogram onto the other by systematic transformation analysis (correct answer)

Question 6

Are the two figures congruent? Which rigid motion(s) justify your answer, based only on the diagram?

Diagram description: A coordinate plane is shown with two polygons.

The coordinate grid has uniform units with labeled axes.
Figure 1 is triangle $\triangle A B C$ with points $A(-4,1)$ , $B(-2,3)$ , and $C(-1,0)$ .
Figure 2 is triangle $\triangle D E F$ with points $D(4,1)$ , $E(2,3)$ , and $F(1,0)$ .
No side lengths or angle measures are marked beyond what can be inferred from the coordinates. Diagram not drawn to scale beyond the uniform grid.

Yes; reflecting Figure 1 across the $y$ -axis maps it onto Figure 2. (correct answer)
No; the triangles are on opposite sides of the $y$ -axis, so they cannot be congruent.
Yes; dilating Figure 1 by a factor of $-1$ maps it onto Figure 2.
Yes; translating Figure 1 up 2 units maps it onto Figure 2.

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Geometry Help: Congruence Via Rigid Motions

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Geometry Help: Congruence Via Rigid Motions

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