Question

An illustration of a coordinate plane of 4 quadrants shows x and y axes ranging from negative 8 to 8 in increments of 1. A triangle upper A, upper B, upper C is formed by solid lines at upper A left parenthesis 2 comma 1 right parenthesis, upper B left parenthesis 7 comma 1 right parenthesis, and upper C left parenthesis 4 comma 3 right parenthesis. A triangle upper A prime, upper B prime, upper C prime is formed by dotted lines at upper A prime left parenthesis negative 1 comma 2 right parenthesis, upper B prime left parenthesis negative 1 comma 7 right parenthesis, upper C prime left parenthesis negative 3 comma 4 right parenthesis.

which transformation shows congruence between triangle abc and triangle a'b'c'

__-degree counter clockwise rotation

Answer 1

To determine the transformation that shows congruence between triangle ABC (with vertices A(2, 1), B(7, 1), C(4, 3)) and triangle A'B'C' (with vertices A'(-1, 2), B'(-1, 7), C'(-3, 4)), we can analyze the coordinates of both triangles.

1. **Locate Triangle ABC**:
- A(2, 1)
- B(7, 1)
- C(4, 3)

2. **Locate Triangle A'B'C'**:
- A'(-1, 2)
- B'(-1, 7)
- C'(-3, 4)

3. **Determine the Relationship**:
- Triangle ABC is positioned in the first quadrant while triangle A'B'C' is positioned in the second quadrant.
- We can use a combination of transformations (rotations and reflections) and check if either achieve congruence.

4. **Testing Rotations**:
- A 90-degree counterclockwise rotation around the origin transforms point (x, y) to (-y, x).
- Applying a 90-degree counterclockwise rotation to the vertices of triangle ABC:
- A(2, 1) transforms to (-1, 2),
- B(7, 1) transforms to (-1, 7),
- C(4, 3) transforms to (-3, 4).

5. **Results**:
After this transformation, we achieve the vertices of triangle A'B'C'.

So, the transformation that shows congruence between triangle ABC and triangle A'B'C' is indeed a **90-degree counterclockwise rotation** around the origin.