I'm stuck!! Can someone please help me with this problem??
A triangle has vertices (1, 4), (1, 1), and (-3, 1). The triangle is dilated by a scale factor of 2, then translated 5 units up, and then rotated 90 degrees counterclockwise about the origin. What are the vertices of the image of the triangle?
dilate by 2: (x,y) -> (2x,2y)
up 5: (x,y) -> (x,y+5)
rotate: (x,y) -> (-y,x)
combining them,
(x,y) -> (2x,2y) -> (2x,2y+5) -> (-(2y+5),2x)
so,
(1,4) -> (-13,2)
and so on for the other points
(1,4) dilated by -1, then rotated 90 degrees counter clock wise
To find the image of the triangle after the given transformations, we need to follow these steps:
1. Dilate the triangle by a scale factor of 2.
2. Translate the triangle 5 units up.
3. Rotate the triangle 90 degrees counterclockwise about the origin.
Let's start with the first step:
1. Dilate the triangle by a scale factor of 2:
- Multiply the coordinates of each vertex by the scale factor.
- The new coordinates of the vertices will be:
- Vertex 1: (2 * 1, 2 * 4) = (2, 8)
- Vertex 2: (2 * 1, 2 * 1) = (2, 2)
- Vertex 3: (2 * -3, 2 * 1) = (-6, 2)
Now let's move on to the second step:
2. Translate the triangle 5 units up:
- Add 5 to the y-coordinate of each vertex.
- The new coordinates of the vertices will be:
- Vertex 1: (2, 8 + 5) = (2, 13)
- Vertex 2: (2, 2 + 5) = (2, 7)
- Vertex 3: (-6, 2 + 5) = (-6, 7)
Lastly, we'll perform the third step:
3. Rotate the triangle 90 degrees counterclockwise about the origin:
- Swap the x and y coordinates of each vertex and negate the new x-coordinate.
- The new coordinates of the vertices will be:
- Vertex 1: (13, -2)
- Vertex 2: (7, -2)
- Vertex 3: (7, 6)
Therefore, the vertices of the image triangle after the given transformations are (13, -2), (7, -2), and (7, 6).
To find the vertices of the image of the triangle after the given transformations, we need to follow these steps:
1. Dilate the triangle by a scale factor of 2:
- Multiply the x-coordinates and y-coordinates of each vertex by 2.
- For the original triangle's vertices:
- (1, 4) becomes (2, 8)
- (1, 1) becomes (2, 2)
- (-3, 1) becomes (-6, 2)
2. Translate the triangle 5 units up:
- Add 5 to the y-coordinates of each vertex.
- For the dilated triangle's vertices:
- (2, 8) becomes (2, 13)
- (2, 2) becomes (2, 7)
- (-6, 2) becomes (-6, 7)
3. Rotate the triangle 90 degrees counterclockwise about the origin:
- Swap the x and y-coordinates of each vertex and negate the new x-coordinate.
- For the translated triangle's vertices:
- (2, 13) becomes (13, -2)
- (2, 7) becomes (7, -2)
- (-6, 7) becomes (7, 6)
Therefore, the vertices of the image of the triangle are:
(13, -2), (7, -2), and (7, 6).