On a coordinate plane, 5 parallelograms are shown. Parallelogram L M N P has points (3, 1), (4, 1), (4, 3), and (5, 3). Figure D is reflected across the x-axis. Figure B is reflected across the y-axis. Figure C is rotated in the first quadrant. Figure A is rotated in the third quadrant.

Question

Which figure represents the image of parallelogram LMNP after a reflection across the line y = x?

figure A
figure B
figure C
figure D

Answer 1

To determine which figure represents the image of parallelogram LMNP after a reflection across the line \( y = x \), we need to apply the reflection transformation to the coordinates of LMNP.

The vertices of parallelogram LMNP are:

• L (3, 1)
• M (4, 1)
• N (4, 3)
• P (5, 3)

Reflecting a point (x, y) across the line \( y = x \) results in the point (y, x). Therefore, we will apply this transformation to each vertex:

1. \( L (3, 1) \rightarrow (1, 3) \)
2. \( M (4, 1) \rightarrow (1, 4) \)
3. \( N (4, 3) \rightarrow (3, 4) \)
4. \( P (5, 3) \rightarrow (3, 5) \)

Now we have the new vertices after reflection:

• L' (1, 3)
• M' (1, 4)
• N' (3, 4)
• P' (3, 5)

Next, we need to check which of the given figures (A, B, C, D) corresponds to these new coordinates.

1. Figure A: Check if it has the points (1, 3), (1, 4), (3, 4), (3, 5).
2. Figure B: Check if it has the points that correspond to the new vertices.
3. Figure C: Check if it has the points that correspond to the new vertices.
4. Figure D: Check if it has the points that correspond to the new vertices.

Without visually inspecting each figure, one must verify the vertices' coordinates.

However, based on the information given, if the figures are distinctly labeled and corresponding points are identifiable, you would typically check each figure methodically against the derived coordinates to identify the correct image after the transformation.

If you're following along, please inspect each figure to see which one has the coordinates:

• (1, 3)
• (1, 4)
• (3, 4)
• (3, 5)

The correct answer will be the figure that corresponds to those coordinates. If you can identify that, you will know which figure represents the reflection across the line \( y = x \).