Identify the rotation rule on a coordinate plane that verifies that triangle A(2,-1), B(4,1), C(3,3) and triangle A'(1,2), B'(-1,4), C'(-3,3) are congruent when rotated 90°.
The rotation rule that verifies the congruence of the triangles is:
(x, y) → (-y, x)
Applying this rule to triangle A(2,-1), we get:
A'(-1, 2)
Applying the rule to triangle B(4, 1), we get:
B'(-1, 4)
Applying the rule to triangle C(3, 3), we get:
C'(-3, 3)
Therefore, triangle A'(-1, 2), B'(-1, 4), and C'(-3, 3) are congruent to triangle A(2,-1), B(4,1), and C(3,3) when rotated 90°.
To identify the rotation rule that verifies the congruence of the given triangles, we need to find the relationship between the corresponding vertices after a rotation of 90 degrees.
First, let's consider the rotation of a point (x, y) by 90 degrees counterclockwise on the coordinate plane.
The rotation rule for a counterclockwise rotation of 90 degrees is as follows:
(x, y) → (-y, x)
Now, let's apply this rule to the vertices of triangle A(2, -1), B(4, 1), C(3, 3) to find the corresponding vertices after a 90-degree counterclockwise rotation.
For point A(2, -1):
After rotating 90 degrees counterclockwise, we have A'(-(-1), 2) = A'(1, 2)
Similarly, for point B(4, 1):
After rotating 90 degrees counterclockwise, we have B'(-1, 4)
And for point C(3, 3):
After rotating 90 degrees counterclockwise, we have C'(-3, 3)
Therefore, the rotated points A'(1, 2), B'(-1, 4), C'(-3, 3) correspond to the original points A(2,-1), B(4,1), C(3,3) after a 90-degree counterclockwise rotation.
This implies that triangle A(2,-1), B(4,1), C(3,3) and triangle A'(1,2), B'(-1,4), C'(-3,3) are congruent after a 90-degree counterclockwise rotation.
To identify the rotation rule on a coordinate plane, we can follow a step-by-step process.
Step 1: Draw the original triangle ABC and its corresponding image triangle A'B'C' after rotation.
Step 2: Determine the center of rotation. In this case, since the triangles are congruent after a 90° rotation, the center of rotation is the midpoint of line segment AA' (the midpoint of (2, -1) and (1, 2)).
Step 3: Draw the line connecting the center of rotation to any vertex of the original triangle. In this case, we can draw the line from the center of rotation (1.5, 0.5) to vertex A (2, -1).
Step 4: Determine the angle of rotation. Since the triangles are congruent after a 90° rotation, the angle of rotation is 90°.
Step 5: Determine the direction of rotation. In this case, the triangles are rotated counterclockwise since the image triangle A'B'C' is positioned to the left of the original triangle ABC.
Step 6: Use the distance between the center of rotation and each vertex of the original triangle to find the coordinates of the corresponding vertices in the image triangle.
Following these steps, we can find the coordinates of the image triangle:
- Triangle A'(1, 2):
- x-coordinate: The x-coordinate of the center of rotation (1.5) plus the distance between the center of rotation and vertex A (2.5 - 1.5 = 1) = 1.5 + 1 = 2.5
- y-coordinate: The y-coordinate of the center of rotation (0.5) plus the distance between the center of rotation and vertex A (0.5 + 1 = 1.5)
- Triangle B'(-1, 4):
- x-coordinate: The x-coordinate of the center of rotation (1.5) plus the distance between the center of rotation and vertex B (1.5 - 1.5 = 0) = 1.5 + 0 = 1.5
- y-coordinate: The y-coordinate of the center of rotation (0.5) plus the distance between the center of rotation and vertex B (0.5 + 2 = 2.5)
- Triangle C'(-3, 3):
- x-coordinate: The x-coordinate of the center of rotation (1.5) plus the distance between the center of rotation and vertex C (0.5 - 1.5 = -1) = 1.5 + (-1) = 0.5
- y-coordinate: The y-coordinate of the center of rotation (0.5) plus the distance between the center of rotation and vertex C (0.5 + 2 = 2.5)
Therefore, the image triangle A'B'C' after a 90° counterclockwise rotation is:
Triangle A'(2.5, 1.5), B'(1.5, 2.5), C'(0.5, 2.5).