Which Triangle Pairs Can Be Mapped Using a Single Translation? Answer

Which of these triangle pairs can be mapped to each other using a single translation? This geometry question asks you to identify which pair of triangles can be moved from one position to another using only a translation — a slide in one direction, with no rotation, reflection, or resizing involved.

Because the original source does not provide labeled diagrams or specific answer choices (A, B, C, D) with visible triangle coordinates, a definitive answer cannot be confirmed without seeing the actual image or diagram. The image below was included with the question but does not render with enough detail to identify the pairs. What follows is a complete explanation of how to solve this type of problem so you can determine the correct answer yourself once you view the diagram.

What Makes a Transformation a Translation?

A translation is the simplest of the four rigid transformations in geometry. It slides every point of a figure the same distance in the same direction. Think of it as picking up a shape and placing it somewhere else without turning it or flipping it. Three properties stay identical between the original triangle and its image after a translation: side lengths, angle measures, and orientation (the order of vertices, clockwise or counterclockwise, does not reverse).

This last property — preserved orientation — is what separates a translation from a reflection. In a reflection the figure appears mirrored, so the vertex order reverses. In a rotation the figure is turned around a fixed point, so the sides point in different directions even though orientation is preserved. A translation changes nothing except position.

How to Identify the Correct Pair in the Diagram

When you look at the answer choices, apply this quick three-step check to each triangle pair:

1. Same size and shape? Measure or compare corresponding sides. If one triangle is larger or has different angles, the mapping is not a rigid motion at all — eliminate that pair immediately.
2. Same orientation? Label the vertices of each triangle in order (e.g., going clockwise). If one triangle’s vertices run clockwise while the other’s run counterclockwise, the mapping requires a reflection, not a translation. Eliminate that pair.
3. Parallel corresponding sides? In a pure translation every side of the image triangle is parallel to the corresponding side of the original. If any pair of corresponding sides is not parallel, the mapping involves a rotation. Eliminate that pair.

The pair that passes all three checks — congruent, same orientation, and all corresponding sides parallel — is the one that can be mapped using a single translation. You can verify by drawing a vector from any vertex of the first triangle to the corresponding vertex of the second; every other vertex pair should share that exact same vector.

Why Other Pairs Typically Fail

In problems like this, distractors are carefully designed. The most common traps are pairs that involve a glide reflection — a combination of a translation and a reflection. These pairs look shifted, so at first glance they seem translated, but one triangle is actually a mirror image of the other. Check orientation to catch this. Another frequent distractor is a pair related by a rotation of 180°. Because a half-turn keeps the figure looking “upright” in some cases, students mistake it for a slide. Verify by checking whether corresponding sides are truly parallel; in a 180° rotation they are parallel, but the direction along each side is reversed, which means vertices map in the opposite order around the triangle. A pair related by a smaller rotation (e.g., 90°) is usually easier to spot because the triangles clearly face different directions.

If two triangles are different sizes, the mapping is a dilation or a similarity transformation — never a translation. This is the easiest distractor to rule out.

Tips to Quickly Recognize Translation Problems

Whenever a question asks about a “single translation,” train your eye to look for the pair that appears copy-pasted in a new location — same tilt, same flip, same size, just shifted. A reliable shortcut: pick one distinctive vertex (say, the sharpest angle) on each triangle and mentally draw an arrow between them. Then check whether the same arrow works for the other two vertex pairs. If all three arrows are identical in length and direction, you have a translation. If even one arrow differs, the transformation is something else. Practicing this arrow test on graph paper builds speed for timed exams.