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 1, negative 3), (1, 1), (2, negative 5). A blue triangle has the following vertices: (negative 4, 1), (negative 2, 5), (negative 1, negative 1).
Question content area bottom
Part 1
The translation rule is (x,y)right arrow(xplus(
negative 3),yplus(
4)).
-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 1, negative 3), (1, 1), (2, negative 5). A blue triangle has the following vertices: (negative 4, 1), (negative 2, 5), (negative 1, negative 1).
Question content area bottom
Part 1
The translation rule is (x,y)right arrow(xplus(
negative 3),yplus(
4)).
Answers
GPT-4o mini
To describe the translation from the black triangle to the blue triangle, we can determine the shift by comparing the coordinates of corresponding vertices.
1. **Black triangle vertices**:
- A: (-1, -3)
- B: (1, 1)
- C: (2, -5)
2. **Blue triangle vertices**:
- A': (-4, 1)
- B': (-2, 5)
- C': (-1, -1)
Now, let's calculate the translation for each vertex:
- For vertex A to A':
- From (-1, -3) to (-4, 1)
- Change in x: -4 - (-1) = -4 + 1 = -3
- Change in y: 1 - (-3) = 1 + 3 = 4
- For vertex B to B':
- From (1, 1) to (-2, 5)
- Change in x: -2 - 1 = -3
- Change in y: 5 - 1 = 4
- For vertex C to C':
- From (2, -5) to (-1, -1)
- Change in x: -1 - 2 = -3
- Change in y: -1 - (-5) = -1 + 5 = 4
In all cases, we can see that the change is consistent:
- **Change in x**: -3
- **Change in y**: +4
Thus, the translation rule can be expressed as:
\[
(x, y) \rightarrow (x - 3, y + 4)
\]
This confirms that the correct translation rule is indeed:
\[
(x, y) \rightarrow (x + (-3), y + 4).
\]
1. **Black triangle vertices**:
- A: (-1, -3)
- B: (1, 1)
- C: (2, -5)
2. **Blue triangle vertices**:
- A': (-4, 1)
- B': (-2, 5)
- C': (-1, -1)
Now, let's calculate the translation for each vertex:
- For vertex A to A':
- From (-1, -3) to (-4, 1)
- Change in x: -4 - (-1) = -4 + 1 = -3
- Change in y: 1 - (-3) = 1 + 3 = 4
- For vertex B to B':
- From (1, 1) to (-2, 5)
- Change in x: -2 - 1 = -3
- Change in y: 5 - 1 = 4
- For vertex C to C':
- From (2, -5) to (-1, -1)
- Change in x: -1 - 2 = -3
- Change in y: -1 - (-5) = -1 + 5 = 4
In all cases, we can see that the change is consistent:
- **Change in x**: -3
- **Change in y**: +4
Thus, the translation rule can be expressed as:
\[
(x, y) \rightarrow (x - 3, y + 4)
\]
This confirms that the correct translation rule is indeed:
\[
(x, y) \rightarrow (x + (-3), y + 4).
\]