Question
The blue figure is a translation image of the black figure. Write a rule to describe the translation.
-6
-3
3
6
-6
-3
3
6
x
y
A coordinate system has a horizontal x-axis and a vertical y-axis each labeled from negative 6 to 6 in increments of 1. A black triangle has the following vertices: (negative 2, negative 3), (3, 0), (4, negative 4). A blue triangle has the following vertices: (negative 4, 1), (1, 4), (2, 0).
Question content area bottom
Part 1
The translation rule is (x,y)right arrow(xplus(
enter your response here),yplus(
enter your response here)).
-6
-3
3
6
-6
-3
3
6
x
y
A coordinate system has a horizontal x-axis and a vertical y-axis each labeled from negative 6 to 6 in increments of 1. A black triangle has the following vertices: (negative 2, negative 3), (3, 0), (4, negative 4). A blue triangle has the following vertices: (negative 4, 1), (1, 4), (2, 0).
Question content area bottom
Part 1
The translation rule is (x,y)right arrow(xplus(
enter your response here),yplus(
enter your response here)).
Answers
GPT-4o mini
To find the translation rule for the transformation from the black triangle to the blue triangle, we can observe the changes in the coordinates of the corresponding vertices of the two triangles.
The vertices of the black triangle are:
1. \((-2, -3)\)
2. \((3, 0)\)
3. \((4, -4)\)
The vertices of the blue triangle are:
1. \((-4, 1)\)
2. \((1, 4)\)
3. \((2, 0)\)
Now we can calculate the translation by finding how much we added to the x-coordinates and y-coordinates when moving from the black triangle to the blue triangle.
### For the first vertex:
- From \((-2, -3)\) to \((-4, 1)\):
- Change in x: \(-4 - (-2) = -4 + 2 = -2\)
- Change in y: \(1 - (-3) = 1 + 3 = 4\)
### For the second vertex:
- From \((3, 0)\) to \((1, 4)\):
- Change in x: \(1 - 3 = -2\)
- Change in y: \(4 - 0 = 4\)
### For the third vertex:
- From \((4, -4)\) to \((2, 0)\):
- Change in x: \(2 - 4 = -2\)
- Change in y: \(0 - (-4) = 0 + 4 = 4\)
From the above calculations, we see that:
- The change in the x-coordinate is consistently \(-2\).
- The change in the y-coordinate is consistently \(4\).
Thus, the translation rule can be described as:
\[
(x,y) \rightarrow (x - 2, y + 4)
\]
### Final Answer:
**The translation rule is** \((x,y) \rightarrow (x - 2, y + 4)\).
The vertices of the black triangle are:
1. \((-2, -3)\)
2. \((3, 0)\)
3. \((4, -4)\)
The vertices of the blue triangle are:
1. \((-4, 1)\)
2. \((1, 4)\)
3. \((2, 0)\)
Now we can calculate the translation by finding how much we added to the x-coordinates and y-coordinates when moving from the black triangle to the blue triangle.
### For the first vertex:
- From \((-2, -3)\) to \((-4, 1)\):
- Change in x: \(-4 - (-2) = -4 + 2 = -2\)
- Change in y: \(1 - (-3) = 1 + 3 = 4\)
### For the second vertex:
- From \((3, 0)\) to \((1, 4)\):
- Change in x: \(1 - 3 = -2\)
- Change in y: \(4 - 0 = 4\)
### For the third vertex:
- From \((4, -4)\) to \((2, 0)\):
- Change in x: \(2 - 4 = -2\)
- Change in y: \(0 - (-4) = 0 + 4 = 4\)
From the above calculations, we see that:
- The change in the x-coordinate is consistently \(-2\).
- The change in the y-coordinate is consistently \(4\).
Thus, the translation rule can be described as:
\[
(x,y) \rightarrow (x - 2, y + 4)
\]
### Final Answer:
**The translation rule is** \((x,y) \rightarrow (x - 2, y + 4)\).