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° counterclockwise about the origin. What are the vertices of the image of the triangle

1. The new vertices would be (-13,2) and (-7,2) and (-7,-6)

2. x = -1 and y = 2
3. William scored 26 and Porter scored 28

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To find the vertices of the image of the triangle after the given transformations, we can perform each transformation step-by-step.

1. Dilation with a scale factor of 2:
To dilate the triangle with a scale factor of 2, we multiply the coordinates of each vertex by 2.
The new coordinates are:
- Vertex 1: (2 * 1, 2 * 4) = (2, 8)
- Vertex 2: (2 * 1, 2 * 1) = (2, 2)
- Vertex 3: (2 * -3, 2 * 1) = (-6, 2)

2. Translation 5 units up:
To translate the triangle 5 units up, we add 5 to the y-coordinate of each vertex.
The new coordinates are:
- Vertex 1: (2, 8 + 5) = (2, 13)
- Vertex 2: (2, 2 + 5) = (2, 7)
- Vertex 3: (-6, 2 + 5) = (-6, 7)

3. Rotation 90° counterclockwise about the origin:
To rotate the triangle 90° counterclockwise, we can use the following rotation matrix:
| 0 -1 |
| 1 0 |
Applying this matrix to each vertex, we get:
- Vertex 1: (0 * 2 + (-1) * 13, 1 * 2 + 0 * 13) = (-13, 2)
- Vertex 2: (0 * 2 + (-1) * 7, 1 * 2 + 0 * 7) = (-7, 2)
- Vertex 3: (0 * -6 + (-1) * 7, 1 * -6 + 0 * 7) = (-7, -6)

Therefore, the vertices of the image of the triangle after the given transformations are (-13, 2), (-7, 2), and (-7, -6).

To find the image of the triangle after the given transformations, we need to apply them in the following order: dilation, translation, and rotation.

1. Dilation:
A dilation with a scale factor of 2 means that each coordinate of the triangle will be multiplied by 2. Thus, the new coordinates will be:
(1*2, 4*2) = (2, 8)
(1*2, 1*2) = (2, 2)
(-3*2, 1*2) = (-6, 2)

2. Translation:
To translate the triangle 5 units up, we need to add 5 to the y-coordinate of each point. The new coordinates become:
(2, 8) + (0, 5) = (2, 13)
(2, 2) + (0, 5) = (2, 7)
(-6, 2) + (0, 5) = (-6, 7)

3. Rotation:
To rotate the triangle 90 degrees counterclockwise about the origin, we can swap the x and y coordinates and negate the new x-coordinate. The new coordinates are:
(13, 2)
(7, 2)
(7, -6)

Therefore, the vertices of the image of the triangle after the given transformations are: (13, 2), (7, 2), and (7, -6).

dilation takes (x,y)->(2x,2y)

translation takes (x,y)->(x,y+5)
rotation takes (x,y)->(-y,x)

So, combining, we get

(x,y) -> (2x,2y) -> (2x,2y+5) -> (-2y-5,2x)

So, just apply that to each coordinate. For example,

(4,3)-> (-11,8)