Question
19 of 1919 of 19 Items
Question
a. What two transformations were completed to move triangle ABC to triangle XYZ? Include the name of the transformation and by how much. Example- it was Rotated by 180 degrees and reflected over line x=y
b. Are the shapes congruent? How do you know?
Question
a. What two transformations were completed to move triangle ABC to triangle XYZ? Include the name of the transformation and by how much. Example- it was Rotated by 180 degrees and reflected over line x=y
b. Are the shapes congruent? How do you know?
Answers
GPT-5 mini
I can do this, but I need the picture or coordinates of triangles ABC and XYZ (or a clear description of where each vertex is). Without that I can only tell you how to find the two transformations and how to decide congruence.
Steps you can follow (or send me the coordinates/image and I’ll do them for you):
1. Identify corresponding vertices (A↔X, B↔Y, C↔Z). If you’re not sure which corresponds to which, compare side lengths or orientation to match them.
2. Check side lengths to see if the triangles are the same size:
- Compute lengths AB, BC, CA and XY, YZ, ZX.
- If all three pairs of corresponding lengths are equal, the triangles are congruent under rigid motions (no dilation).
3. Determine the transformations:
- Translation: compute vector v = (xX − xA, yX − yA). If adding that vector to A, B, C gives X, Y, Z, then the map is a translation by v.
- Rotation: if a rotation about a point O maps A→X, compute angle and direction by measuring ∠AOX. You can compute the angle from vectors OA and OX using dot product or by comparing slopes; direction is clockwise or counterclockwise.
- Reflection: if there’s a line m such that the perpendicular bisector of segment AX (and BY, CZ) is m, then it’s a reflection across line m. Find the equation of that line.
- Combination: often you’ll have a translation then a rotation/reflection (or vice versa). Apply one transformation to check if the resulting intermediate matches the target.
4. Orientation test (to tell rotation vs reflection order):
- If orientation (clockwise vs counterclockwise ordering of vertices) is preserved, then the move can be achieved by translations and rotations only.
- If orientation is reversed, a reflection (or a rotation+reflection) is involved.
How to describe the answer (examples):
- “Translated by vector <3, −2>, then rotated 90° counterclockwise about the origin.”
- “Reflected across the line y = x, then translated left 4.”
- If congruent: “Yes — they are congruent, because ABC can be carried onto XYZ by the two rigid motions above (or because corresponding side lengths/angles are equal).”
- If not congruent: “No — a dilation (scale factor ≠ 1) is needed, so they are not congruent.”
If you paste the coordinate list or upload the diagram, I’ll compute the exact transformations and give the final answer.
Steps you can follow (or send me the coordinates/image and I’ll do them for you):
1. Identify corresponding vertices (A↔X, B↔Y, C↔Z). If you’re not sure which corresponds to which, compare side lengths or orientation to match them.
2. Check side lengths to see if the triangles are the same size:
- Compute lengths AB, BC, CA and XY, YZ, ZX.
- If all three pairs of corresponding lengths are equal, the triangles are congruent under rigid motions (no dilation).
3. Determine the transformations:
- Translation: compute vector v = (xX − xA, yX − yA). If adding that vector to A, B, C gives X, Y, Z, then the map is a translation by v.
- Rotation: if a rotation about a point O maps A→X, compute angle and direction by measuring ∠AOX. You can compute the angle from vectors OA and OX using dot product or by comparing slopes; direction is clockwise or counterclockwise.
- Reflection: if there’s a line m such that the perpendicular bisector of segment AX (and BY, CZ) is m, then it’s a reflection across line m. Find the equation of that line.
- Combination: often you’ll have a translation then a rotation/reflection (or vice versa). Apply one transformation to check if the resulting intermediate matches the target.
4. Orientation test (to tell rotation vs reflection order):
- If orientation (clockwise vs counterclockwise ordering of vertices) is preserved, then the move can be achieved by translations and rotations only.
- If orientation is reversed, a reflection (or a rotation+reflection) is involved.
How to describe the answer (examples):
- “Translated by vector <3, −2>, then rotated 90° counterclockwise about the origin.”
- “Reflected across the line y = x, then translated left 4.”
- If congruent: “Yes — they are congruent, because ABC can be carried onto XYZ by the two rigid motions above (or because corresponding side lengths/angles are equal).”
- If not congruent: “No — a dilation (scale factor ≠ 1) is needed, so they are not congruent.”
If you paste the coordinate list or upload the diagram, I’ll compute the exact transformations and give the final answer.