What is the sequence of steps used to move the triangle in quadrant four to the triangle in quadrant two to show a that the triangles are congruent?
Original position A (-3, 4) B (-4, 2) C (-2, 2)
New position A (2, 3) B (1, 1) C (3, 1)
The options are
A rotation, then translation
B reflection, then translation
C translation then reflection
D translation, then transformation
A rotation, then translation
To move the triangle in quadrant four to the triangle in quadrant two and show that they are congruent, we need to identify the correct sequence of steps. Let's analyze the given options:
A) Rotation, then translation: This option involves rotating the triangle and then performing a translation. However, this does not match the given coordinates because the triangle in quadrant four has different vertex positions from the triangle in quadrant two. Therefore, option A is incorrect.
B) Reflection, then translation: This option involves reflecting the triangle and then performing a translation. Let's check if this matches the given coordinates:
1. Reflect the triangle across the x-axis: This will change the y-coordinates of the vertices to their negative counterparts. The reflected coordinates are: A (-3, -4), B (-4, -2), C (-2, -2).
2. Perform a translation: Now, we translate the reflected triangle to match the new position. Shifting each vertex by (5, -1), the new coordinates become: A (-3 + 5, -4 - 1) = (2, -5), B (-4 + 5, -2 - 1) = (1, -3), C (-2 + 5, -2 - 1) = (3, -3).
The resulting coordinates are not the same as the new position given (A (2, 3) B (1, 1) C (3, 1)). Therefore, option B is incorrect.
C) Translation, then reflection: This option involves performing a translation and then reflecting the triangle. Let's check if this matches the given coordinates:
1. Perform a translation: We translate the original triangle by shifting each vertex by (5, -1):
- A (-3 + 5, 4 - 1) = (2, 3)
- B (-4 + 5, 2 - 1) = (1, 1)
- C (-2 + 5, 2 - 1) = (3, 1)
2. Reflect the translated triangle across the x-axis: This will change the y-coordinates of the vertices to their negative counterparts. The reflected coordinates are:
- A (2, -3)
- B (1, -1)
- C (3, -1)
The resulting coordinates match the new position given (A (2, 3) B (1, 1) C (3, 1)). Therefore, option C is correct.
D) Translation, then transformation: This option does not specify the type of transformation, so it is not clear how it would be applied. Therefore, option D is incorrect.
In conclusion, the correct sequence of steps to move the triangle in quadrant four to the triangle in quadrant two and show that they are congruent is:
C) Translation, then reflection.