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All courses Math I · G-CO.6 41 of 59
Use rigid motions to transform figures and decide whether two figures are congruent.

Decide congruence by verifying a translation

Problem
Are these figures congruent by a translation? Compare preimage vertices 0, 0; 3, 0; 0, 2 with image vertices 5, -1; 8, -1; 5, 1.
Source and image triangles: (0,0),(3,0),(0,2) correspond to (5,-1),(8,-1),(5,1). Open full size
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Hint

Compare one preimage vertex to its image to find the translation vector, then test whether the same vector works for the other vertices.

Figures are congruent by a translation only if every corresponding vertex moves by one common vector.

Solution walkthrough

01

Compute a candidate translation

\[(0,0)~\text{to}~(5,-1)~\text{gives}~\text{vector}~(5,-1)\]

The first corresponding pair moves five units right and one unit down.

02

Verify the remaining pairs

\[(3,0)+(5,-1)=(8,-1);~(0,2)+(5,-1)=(5,1)\]

The same vector maps both remaining preimage vertices to the listed image vertices.

03

Use the rigid-motion consequence

\[\text{one}~\text{translation}~\text{maps}~\text{the}~\text{entire}~\text{triangle};~\text{distances}~\text{and}~\text{angles}~\text{are}~\text{preserved}\]

A translation is a rigid motion, so an exact vertex match proves the figures have equal size and shape.

04

State the congruence conclusion

\[\text{Yes},~\text{the}~\text{figures}~\text{are}~\text{congruent}~\text{because}~\text{each}~\text{vertex}~\text{moves}~\text{by}~\text{the}~\text{same}~\text{translation}~\text{vector}~⟨5,-1⟩.\]

All three checked correspondences use the single vector required for congruence by translation.

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Another way

  1. Observe that adding (5,-1) to the entire inspected preimage triangle makes it coincide with the image triangle.

!

Common mistake

Do not use a vector between two vertices of the same triangle. Compare each preimage vertex with its designated image vertex.

Worked result: Every corresponding vertex moves by ⟨5,-1⟩, so one translation maps the first triangle onto the second and they are congruent.
congruent: yes; translation: <5,-1>