Congruence via Rigid Motions - Geometry

1. 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?
   Answer: A rotation of $70°$ about point $O$, because the composition of reflections across intersecting lines creates double the intersection angle 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.
2. 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 $MNO$?
   Answer: They are congruent because any sequence of rigid motions connecting two figures through intermediates establishes congruence between endpoints 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.
3. 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?
   Answer: Translation by vector $(5, 12)$, because moving from one center to another with same orientation requires only displacement 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.
4. 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?
   Answer: The hexagons are congruent because rigid motions preserve shape and size regardless of the number of transformations used 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.
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?
   Answer: Determine if there exists a sequence of rigid motions mapping one parallelogram onto the other by systematic transformation analysis 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.
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.
   Answer: Yes; reflecting Figure 1 across the $y$-axis maps it onto Figure 2. 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.

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