Question

Determining Congruence
Given a pre-image and an image, explain how you could use transformations to show that they are congruent.

Answer 1

Two figures are congruent exactly when one can be carried onto the other by rigid motions (isometries): translations, rotations and reflections (or a composition of them). To show congruence with transformations, give an explicit sequence of such motions that sends the pre-image onto the image.

A simple procedure you can follow
1. Choose a convenient pair of corresponding points (usually vertices). Translate the pre-image so that that point lands on its corresponding point in the image.
2. Rotate about that common point so a second corresponding point lines up (this also aligns the segment joining the two points).
3. If after step 2 the remaining points do not coincide, the orientation may be reversed; reflect across the appropriate line (for example the line through two matched points or the perpendicular bisector) so all remaining corresponding points match.

If after these steps every vertex of the pre-image coincides with the corresponding vertex of the image, you have produced an isometry and therefore proven the figures are congruent.

Quick example (triangles ABC → A′B′C′)
- Translate A to A′.
- Rotate about A′ until the image of B lies on the line A′B′; check it lands at B′.
- If C does not land at C′, reflect across the line A′B′ (or another appropriate line). If now C lands at C′, the triangles are congruent.

Alternative approaches
- Show algebraically in coordinates: compute the translation vector and rotation matrix (or reflection) needed to map each vertex and verify it works.
- Or use congruence criteria for polygons (triangles: SSS, SAS, ASA) to conclude existence of a rigid motion.

Any explicit sequence of translations, rotations and/or reflections that maps the pre-image onto the image is a constructive proof of congruence.