Describe the series of transformations that occur have occurred to move the triangle and quadrant 4 to the each triangle in quadrant two to show that the triangles are congruent.
Original position was G (-5, 5) E (-1, 3) H (-5, 3)
New position is G (3, 3) E (7, 1) H (3, 1)
The options are
A reflections then translation
B translation, then rotation
C rotation then reflection
D rotation, then translation
The series of transformations that have occurred to move the triangle in quadrant 4 to the triangle in quadrant 2 are:
D) rotation, then translation
Explanation:
Step 1: Rotation
The first transformation that occurred is a rotation. The triangle is rotated 180 degrees about the origin (0,0). This moves the original triangle in quadrant 4 to quadrant 2. The new coordinates after rotation are:
G' (-5, -5)
E' (-1, -3)
H' (-5, -3)
Step 2: Translation
The second transformation is a translation. The triangle is moved 8 units in the positive x-direction and 2 units in the positive y-direction. This gives us the final new position of the triangle in quadrant 2:
G (3, 3)
E (7, 1)
H (3, 1)
Therefore, the correct answer is D) rotation, then translation.
To show that the triangles are congruent, let's describe the series of transformations that occurred to move the original triangle in quadrant 4 to the new triangle in quadrant 2.
The original triangle has vertices G(-5, 5), E(-1, 3), and H(-5, 3).
The new triangle has vertices G(3, 3), E(7, 1), and H(3, 1).
1. Translation: The initial translation occurs by moving each vertex 8 units to the right and 2 units down.
G(-5, 5) -> G(3, 3)
E(-1, 3) -> E(7, 1)
H(-5, 3) -> H(3, 1)
2. Reflection: After the translation, a reflection occurs about the y-axis.
G(3, 3) -> G(-3, 3)
E(7, 1) -> E(-7, 1)
H(3, 1) -> H(-3, 1)
Now, if we compare the resulting triangle G(-3, 3), E(-7, 1), and H(-3, 1) (after the reflection), with the original triangle in quadrant 2, we can see that they are congruent.
Therefore, the series of transformations that occurred to move the original triangle in quadrant 4 to the new triangle in quadrant 2 is: A) reflections then translation.