Question

Which rigid transformations would map ΔJKL onto ΔPQR? Select the three correct answers.

a reflection only
a rotation only
a rotation and a reflection
a translation and a reflection
a translation only

Answer 1

I don’t have the diagram or coordinates, so I can’t pick the three correct answers yet. Tell me the picture (an image) or the coordinates/labels of J,K,L and P,Q,R and I’ll pick the three that work.

Meanwhile, here’s a quick checklist you can use to decide which rigid motions will map ΔJKL onto ΔPQR:

1. Orientation test (decides whether a reflection is needed)
- Compute the signed area (or use the vertex order) of each triangle. If the orientations are the same (both clockwise or both counterclockwise) then no reflection is required; if they are opposite, some reflection must be part of the mapping.

2. Translation only
- Check the vectors JP, KQ, and LR. If they’re all equal (same direction and length), a single translation maps ΔJKL to ΔPQR.

3. Rotation only
- Find a point O (candidate center) such that distances OJ = OP, OK = OQ, OL = OR and the oriented angles ∠JOK = ∠POQ (same rotation angle). Practically: find the intersection of perpendicular bisectors of J–P and K–Q; if that intersection is the same for L–R and the rotation angle from each vertex pair is equal, a single rotation works.

4. Reflection only
- Find the perpendicular bisector of J–P. If that same line also perpendicularly bisects K–Q and L–R (i.e., all three midpoint segments are symmetric across one line), then a single reflection maps the triangles.

5. Rotation + reflection or Translation + reflection (or glide reflection)
- If orientation is reversed but there is no single reflection axis, the mapping may be a composition: a rotation followed by a reflection, or a reflection plus a translation (a glide reflection). You can try to factor a found isometry into these components once you know the correspondence.

If you post the diagram or coordinates I’ll apply these tests and give the three correct choices.