ΔABC with vertices A(-3, 0), B(-2, 3), C(-1, 1) is rotated 180° clockwise about the origin. It is then reflected across the line y = -x. What are the coordinates of the vertices of the image?
rotation: (x,y) → (-x,-y)
reflection: (x,y) → (-y,-x)
so the combination takes (x,y)→(-x,-y)→(y,x)
To solve this problem, let's break it down into two steps: rotating the triangle and then reflecting it.
Step 1: Rotating the Triangle
To rotate the triangle 180° clockwise about the origin, we need to find the new coordinates of each vertex after the rotation.
To rotate a point (x, y) 180° clockwise about the origin, we can use the following formulas:
New X-coordinate = -x,
New Y-coordinate = -y.
Let's apply these formulas to each vertex of the triangle:
For Vertex A(-3, 0) after rotating:
New X-coordinate = -(-3) = 3,
New Y-coordinate = -(0) = 0.
Therefore, the new coordinates of A are A'(3, 0).
For Vertex B(-2, 3) after rotating:
New X-coordinate = -(-2) = 2,
New Y-coordinate = -(3) = -3.
Therefore, the new coordinates of B are B'(2, -3).
For Vertex C(-1, 1) after rotating:
New X-coordinate = -(-1) = 1,
New Y-coordinate = -(1) = -1.
Therefore, the new coordinates of C are C'(1, -1).
Now we have the triangle after the rotation: A'(3, 0), B'(2, -3), C'(1, -1).
Step 2: Reflecting the Triangle across the line y = -x
To reflect a point across the line y = -x, we need to swap the x and y coordinates of the given point.
Applying this reflection to each vertex of the rotated triangle:
For Vertex A'(3, 0) after reflection:
New X-coordinate = 0, New Y-coordinate = 3.
Therefore, the new coordinates of A' after reflection are A''(0, 3).
For Vertex B'(2, -3) after reflection:
New X-coordinate = -3, New Y-coordinate = 2.
Therefore, the new coordinates of B' after reflection are B''(-3, 2).
For Vertex C'(1, -1) after reflection:
New X-coordinate = -1, New Y-coordinate = 1.
Therefore, the new coordinates of C' after reflection are C''(-1, 1).
So, the final coordinates of the vertices of the image after both rotation and reflection are:
A''(0, 3), B''(-3, 2), C''(-1, 1).