The correct transformation mapping ∆abc onto ∆a'b'c' is a rotation 90° clockwise about the origin. The coordinates transformation rules confirm that each vertex of ∆abc is rotated accordingly to match the vertices of ∆a'b'c'.
To determine the transformation that maps ∆ABC onto ∆A'B'C', let's analyze their relative positions and orientations based on their coordinates.
Given:
Triangle 1: A(-3, 1), B(-1, 2), C(-2, 1)
Triangle 2: A'(-1, -3), B'(-2, -1), C'(-1, -2)
Let's evaluate the transformations:
a. Rotation 90° clockwise about the origin would change the signs of x-coordinates and swap x and y values. This doesn't match the corresponding points from ∆ABC to ∆A'B'C'.
b. Rotation 90° counterclockwise about the origin would again change the signs of x-coordinates and swap x and y values, which also doesn't align the points.
c. Reflection across the x-axis would only change the signs of y-coordinates, but the x-coordinates would remain the same, and that doesn't match the corresponding points.
d. Reflection across the line y = x swaps the x and y values. Let's apply this transformation to see if it maps ∆ABC onto ∆A'B'C':
For ∆ABC:
- A(-3, 1) → A'(1, -3)
- B(-1, 2) → B'(2, -1)
- C(-2, 1) → C'(1, -2)
These new coordinates indeed match the coordinates of ∆A'B'C', suggesting that the transformation is a reflection across the line y = x, making the correct answer: d. Reflection across the line y = x.
Graph from question: