Question

Triangle P Q R has vertices P negative 2 comma negative 1, Q negative 1 comma 4, and R 1 comma 3. Triangle P prime Q prime R prime has vertices P prime negative 2 comma 1, Q prime negative 1 comma negative 4, and R prime 1 comma negative 3.



Which transformation maps triangle P Q R onto triangle P prime Q prime R prime, preserving side lengths so that the triangles are congruent by S S S?

Answer options with 5 options
A.
reflection over the x-axis

B.
reflection over the y-axis

C.
reflection over the line y equals x

D.
rotation 90° clockwise about the origin

E.
rotation 180° clockwise about the origin

Answer 1

To determine which transformation maps triangle PQR onto triangle P'Q'R' while preserving side lengths (making the triangles congruent by SSS), we need to compare the coordinates of the vertices of the triangles.

The vertices of triangle PQR are:
- P(-2, -1)
- Q(-1, 4)
- R(1, 3)

The vertices of triangle P'Q'R' are:
- P'(-2, 1)
- Q'(-1, -4)
- R'(1, -3)

Now, let’s analyze the transformations one by one to identify which one maps triangle PQR onto triangle P'Q'R':

A. **Reflection over the x-axis**:
- P(-2, -1) becomes P'(-2, 1) (this matches)
- Q(-1, 4) becomes Q'(-1, -4) (this matches)
- R(1, 3) becomes R'(1, -3) (this matches)

All points match after reflection over the x-axis, so this transformation works.

B. **Reflection over the y-axis**:
- P(-2, -1) would become P'(2, -1), which does not match.

This does not work.

C. **Reflection over the line y = x**:
- P(-2, -1) becomes P'(-1, -2), which does not match.

This does not work.

D. **Rotation 90° clockwise about the origin**:
- P(-2, -1) becomes P'(-1, 2), which does not match.

This does not work.

E. **Rotation 180° clockwise about the origin**:
- P(-2, -1) becomes P'(2, 1), which does not match.

This does not work.

The only transformation that maps triangle PQR to triangle P'Q'R' while preserving side lengths, and thus making them congruent by SSS, is **A. reflection over the x-axis**.