Question
You bought a house located at: A (2,3); B( 4,3); C(4,5); D(3,6); E(2,5)
a flood moves your house exactly 4 blocks to the left and two blocks down. Describe specifically the formal transformation that occurred when the flood moved your house. is the new house congruent after translation?
a flood moves your house exactly 4 blocks to the left and two blocks down. Describe specifically the formal transformation that occurred when the flood moved your house. is the new house congruent after translation?
Answers
GPT-4o mini
To analyze the transformation of the house due to the flood, we need to break down the translation that occurred. The flood moved the house 4 blocks to the left and 2 blocks down.
### Formal Transformation
1. **Translation Vector**: The translation can be represented by a vector, which in this case is \((-4, -2)\).
2. **Transformation Description**:
- Moving left means subtracting from the x-coordinate, and moving down means subtracting from the y-coordinate.
- This means for any point \((x, y)\) on the house, the new coordinates \((x', y')\) after translation will be calculated as:
\[
x' = x - 4 \\
y' = y - 2
\]
3. **Applying the Transformation**:
- Let’s apply this transformation to each of your house's corners:
- For Point A (2, 3):
\[
A' = (2 - 4, 3 - 2) = (-2, 1)
\]
- For Point B (4, 3):
\[
B' = (4 - 4, 3 - 2) = (0, 1)
\]
- For Point C (4, 5):
\[
C' = (4 - 4, 5 - 2) = (0, 3)
\]
- For Point D (3, 6):
\[
D' = (3 - 4, 6 - 2) = (-1, 4)
\]
- For Point E (2, 5):
\[
E' = (2 - 4, 5 - 2) = (-2, 3)
\]
### New House Coordinates
After the translation, the new coordinates are:
- A' = (-2, 1)
- B' = (0, 1)
- C' = (0, 3)
- D' = (-1, 4)
- E' = (-2, 3)
### Congruence After Translation
Regarding congruence:
- **Definition of Congruence**: Two figures are congruent if they have the same shape and size.
- **Effect of Translation**: A translation is a rigid transformation, which means it preserves distances and angles between points.
Thus, the new house after translation is congruent to the original house. The shape and size remain unchanged; only the position of the house has changed in the coordinate plane.
### Formal Transformation
1. **Translation Vector**: The translation can be represented by a vector, which in this case is \((-4, -2)\).
2. **Transformation Description**:
- Moving left means subtracting from the x-coordinate, and moving down means subtracting from the y-coordinate.
- This means for any point \((x, y)\) on the house, the new coordinates \((x', y')\) after translation will be calculated as:
\[
x' = x - 4 \\
y' = y - 2
\]
3. **Applying the Transformation**:
- Let’s apply this transformation to each of your house's corners:
- For Point A (2, 3):
\[
A' = (2 - 4, 3 - 2) = (-2, 1)
\]
- For Point B (4, 3):
\[
B' = (4 - 4, 3 - 2) = (0, 1)
\]
- For Point C (4, 5):
\[
C' = (4 - 4, 5 - 2) = (0, 3)
\]
- For Point D (3, 6):
\[
D' = (3 - 4, 6 - 2) = (-1, 4)
\]
- For Point E (2, 5):
\[
E' = (2 - 4, 5 - 2) = (-2, 3)
\]
### New House Coordinates
After the translation, the new coordinates are:
- A' = (-2, 1)
- B' = (0, 1)
- C' = (0, 3)
- D' = (-1, 4)
- E' = (-2, 3)
### Congruence After Translation
Regarding congruence:
- **Definition of Congruence**: Two figures are congruent if they have the same shape and size.
- **Effect of Translation**: A translation is a rigid transformation, which means it preserves distances and angles between points.
Thus, the new house after translation is congruent to the original house. The shape and size remain unchanged; only the position of the house has changed in the coordinate plane.