Question
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)
Original position A (-3, 4) B (-4, 2) C (-2, 2)
New position A (2, 3) B (1, 1) C (3, 1)
Answers
GPT 3.5
To show that the triangles in the original and new positions are congruent, we need to perform a series of translations and rotations. Here are the steps to move the triangle in quadrant four to the triangle in quadrant two:
1) Translate the triangle in the original position to the origin:
- Subtract the x-coordinate of point A (-3) from all x-coordinates: A' (0, 4), B' (-1, 2), C' (1, 2)
2) Translate the triangle at the origin to the new position:
- Add the x-coordinate of point A (2) to all x-coordinates: A'' (2, 4), B'' (1, 2), C'' (3, 2)
3) Translate the triangle at the new position back to its original position in quadrant two:
- Subtract the y-coordinate of point A (1) from all y-coordinates: A''' (2, 3), B''' (1, 1), C''' (3, 1)
4) Finally, perform a rotation of 180 degrees around the origin:
- Multiply the x-coordinates of the points by -1: A'''' (-2, 3), B'''' (-1, 1), C'''' (-3, 1)
Now, the triangle in the original position (-3, 4), (-4, 2), (-2, 2) is congruent to the triangle in the new position (2, 3), (1, 1), (3, 1), by a translation and a 180-degree rotation.
1) Translate the triangle in the original position to the origin:
- Subtract the x-coordinate of point A (-3) from all x-coordinates: A' (0, 4), B' (-1, 2), C' (1, 2)
2) Translate the triangle at the origin to the new position:
- Add the x-coordinate of point A (2) to all x-coordinates: A'' (2, 4), B'' (1, 2), C'' (3, 2)
3) Translate the triangle at the new position back to its original position in quadrant two:
- Subtract the y-coordinate of point A (1) from all y-coordinates: A''' (2, 3), B''' (1, 1), C''' (3, 1)
4) Finally, perform a rotation of 180 degrees around the origin:
- Multiply the x-coordinates of the points by -1: A'''' (-2, 3), B'''' (-1, 1), C'''' (-3, 1)
Now, the triangle in the original position (-3, 4), (-4, 2), (-2, 2) is congruent to the triangle in the new position (2, 3), (1, 1), (3, 1), by a translation and a 180-degree rotation.