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)
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.
To move the triangle in quadrant four to the triangle in quadrant two and show that the triangles are congruent, follow these steps:
1. Identify the corresponding vertices of the triangles. In this case, the corresponding vertices are A, B, and C.
2. Determine the translation vector, which represents the distance and direction to move each vertex. To find the translation vector, subtract the coordinates of the original position from the coordinates of the new position for each vertex.
For vertex A:
New position - Original position
= (2, 3) - (-3, 4)
= (2 + 3, 3 - 4)
= (5, -1)
For vertex B:
New position - Original position
= (1, 1) - (-4, 2)
= (1 + 4, 1 - 2)
= (5, -1)
For vertex C:
New position - Original position
= (3, 1) - (-2, 2)
= (3 + 2, 1 - 2)
= (5, -1)
Therefore, the translation vector for all three vertices is (5, -1).
3. Use the translation vector to move each vertex of the original triangle. Add the translation vector to the coordinates of each vertex.
For vertex A:
New coordinate = Original coordinate + Translation vector
= (-3, 4) + (5, -1)
= (2, 3)
For vertex B:
New coordinate = Original coordinate + Translation vector
= (-4, 2) + (5, -1)
= (1, 1)
For vertex C:
New coordinate = Original coordinate + Translation vector
= (-2, 2) + (5, -1)
= (3, 1)
After moving each vertex using the translation vector, the new position of the triangle will be A (2, 3), B (1, 1), and C (3, 1), which matches the given new position.
4. Finally, compare the corresponding side lengths and angles of the two triangles. If the corresponding side lengths and angles are equal, the triangles are congruent.