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)
In order to show that the triangles in quadrant two are congruent to the original triangle in quadrant four, a series of transformations can be applied.
1. Translation:
The original triangle is translated to the right by 8 units and down by 2 units. This moves point G from (-5, 5) to (3, 3), point E from (-1, 3) to (7, 1), and point H from (-5, 3) to (3, 1).
2. Reflection:
After the translation, a reflection is performed over the y-axis. This flips the triangle horizontally while maintaining the same orientation. As a result, point G is now (-3, 3), point E is now (-7, 1), and point H is now (-3, 1).
3. Rotation:
To align the triangle with quadrant two, a rotation of 180 degrees is applied around the origin. This rotates the triangle counter-clockwise and brings it into the second quadrant. After the rotation, point G becomes (3, 3), point E becomes (7, 1), and point H becomes (3, 1).
By applying these transformations (translation, reflection, and rotation) in sequence, the original triangle can be transformed into the congruent triangle in quadrant two.
To show that the triangles in quadrant two are congruent to the original triangle in quadrant 4, the following transformations have occurred:
1. Translation: A translation of 8 units to the right and 2 units downwards has been applied. This moved the original triangle from quadrant 4 to a new position in quadrant 1. The new vertices of the triangle are:
G' (3, 3)
E' (7, 1)
H' (3, 1)
2. Reflection: The triangle in quadrant 1 has been reflected over the y-axis. This reflects the triangle from quadrant 1 to quadrant 2 while preserving size and shape. The new position of the reflected triangle is:
G'' (-3, 3)
E'' (-7, 1)
H'' (-3, 1)
3. Congruence: After the translation and reflection, the triangle in quadrant 2, G''E''H'', is congruent to the original triangle in quadrant 4, GEH. Both triangles have the same size and shape, and their corresponding sides and angles are congruent.