Congruence via Rigid Motions
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Geometry › Congruence via Rigid Motions
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.
No; the triangles are on opposite sides of the $y$-axis, so they cannot be congruent.
Yes; translating Figure 1 up 2 units maps it onto Figure 2.
Yes; dilating Figure 1 by a factor of $-1$ maps it onto Figure 2.
Yes; reflecting Figure 1 across the $y$-axis 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.