Question
Your house is located at A (2,2), B (4,2), C (4,4), D (3,5), and E (2,4). A flood moves your house exactly 4 blocks to the left and 2 blocks down.Is your new house after this transformation congruent to the original house? Explain
your answer.
your answer.
Answers
GPT-4o mini
To determine if the new house after the flood transformation is congruent to the original house, we first need to find the new coordinates of each point after the flood moves the house exactly 4 blocks to the left and 2 blocks down.
The transformation moves each point (x, y) to (x - 4, y - 2).
We will apply this transformation to each point:
1. **Point A (2, 2)**:
\[
A' = (2 - 4, 2 - 2) = (-2, 0)
\]
2. **Point B (4, 2)**:
\[
B' = (4 - 4, 2 - 2) = (0, 0)
\]
3. **Point C (4, 4)**:
\[
C' = (4 - 4, 4 - 2) = (0, 2)
\]
4. **Point D (3, 5)**:
\[
D' = (3 - 4, 5 - 2) = (-1, 3)
\]
5. **Point E (2, 4)**:
\[
E' = (2 - 4, 4 - 2) = (-2, 2)
\]
Now we have the new coordinates for our points after the flood transformation:
- A' = (-2, 0)
- B' = (0, 0)
- C' = (0, 2)
- D' = (-1, 3)
- E' = (-2, 2)
**Next, we need to check if the two shapes (original and transformed) are congruent.** Two shapes are congruent if they have the same size and shape, which implies that their side lengths and angles are identical, even if they are in different positions.
Thus, we need to calculate the lengths of the sides of the original shape formed by the points A, B, C, D, E and then compare these lengths to the transformed shape.
**Original Points:**
- Length AB: \( \sqrt{(4-2)^2 + (2-2)^2} = \sqrt{2^2} = 2 \)
- Length BC: \( \sqrt{(4-4)^2 + (4-2)^2} = \sqrt{2^2} = 2 \)
- Length CD: \( \sqrt{(3-4)^2 + (5-4)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- Length DE: \( \sqrt{(2-3)^2 + (4-5)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- Length EA: \( \sqrt{(2-2)^2 + (2-4)^2} = \sqrt{2^2} = 2 \)
**Transformed Points:**
- Length A'B': \( \sqrt{(0 - (-2))^2 + (0 - 0)^2} = \sqrt{2^2} = 2 \)
- Length B'C': \( \sqrt{(0 - 0)^2 + (2 - 0)^2} = \sqrt{2^2} = 2 \)
- Length C'D': \( \sqrt{(-1 - 0)^2 + (3 - 2)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- Length D'E': \( \sqrt{(-2 - (-1))^2 + (2 - 3)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- Length E'A': \( \sqrt{(-2 - 2)^2 + (0 - 2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \)
Now we can compare them:
**Original lengths:**
- AB: 2
- BC: 2
- CD: \( \sqrt{2} \)
- DE: \( \sqrt{2} \)
- EA: 2
**Transformed lengths:**
- A'B': 2
- B'C': 2
- C'D': \( \sqrt{2} \)
- D'E': \( \sqrt{2} \)
- E'A': \( 2\sqrt{2} \)
**Conclusion:**
One main side length differs, specifically E'A' in the transformed shape is \( 2\sqrt{2} \) while the original shape had a length of 2 for EA. Therefore, we conclude that the two figures are not congruent since the corresponding side lengths are not the same.
Thus, **the answer is**: **No**, the new house after the transformation is not congruent to the original house.
The transformation moves each point (x, y) to (x - 4, y - 2).
We will apply this transformation to each point:
1. **Point A (2, 2)**:
\[
A' = (2 - 4, 2 - 2) = (-2, 0)
\]
2. **Point B (4, 2)**:
\[
B' = (4 - 4, 2 - 2) = (0, 0)
\]
3. **Point C (4, 4)**:
\[
C' = (4 - 4, 4 - 2) = (0, 2)
\]
4. **Point D (3, 5)**:
\[
D' = (3 - 4, 5 - 2) = (-1, 3)
\]
5. **Point E (2, 4)**:
\[
E' = (2 - 4, 4 - 2) = (-2, 2)
\]
Now we have the new coordinates for our points after the flood transformation:
- A' = (-2, 0)
- B' = (0, 0)
- C' = (0, 2)
- D' = (-1, 3)
- E' = (-2, 2)
**Next, we need to check if the two shapes (original and transformed) are congruent.** Two shapes are congruent if they have the same size and shape, which implies that their side lengths and angles are identical, even if they are in different positions.
Thus, we need to calculate the lengths of the sides of the original shape formed by the points A, B, C, D, E and then compare these lengths to the transformed shape.
**Original Points:**
- Length AB: \( \sqrt{(4-2)^2 + (2-2)^2} = \sqrt{2^2} = 2 \)
- Length BC: \( \sqrt{(4-4)^2 + (4-2)^2} = \sqrt{2^2} = 2 \)
- Length CD: \( \sqrt{(3-4)^2 + (5-4)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- Length DE: \( \sqrt{(2-3)^2 + (4-5)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- Length EA: \( \sqrt{(2-2)^2 + (2-4)^2} = \sqrt{2^2} = 2 \)
**Transformed Points:**
- Length A'B': \( \sqrt{(0 - (-2))^2 + (0 - 0)^2} = \sqrt{2^2} = 2 \)
- Length B'C': \( \sqrt{(0 - 0)^2 + (2 - 0)^2} = \sqrt{2^2} = 2 \)
- Length C'D': \( \sqrt{(-1 - 0)^2 + (3 - 2)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- Length D'E': \( \sqrt{(-2 - (-1))^2 + (2 - 3)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- Length E'A': \( \sqrt{(-2 - 2)^2 + (0 - 2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \)
Now we can compare them:
**Original lengths:**
- AB: 2
- BC: 2
- CD: \( \sqrt{2} \)
- DE: \( \sqrt{2} \)
- EA: 2
**Transformed lengths:**
- A'B': 2
- B'C': 2
- C'D': \( \sqrt{2} \)
- D'E': \( \sqrt{2} \)
- E'A': \( 2\sqrt{2} \)
**Conclusion:**
One main side length differs, specifically E'A' in the transformed shape is \( 2\sqrt{2} \) while the original shape had a length of 2 for EA. Therefore, we conclude that the two figures are not congruent since the corresponding side lengths are not the same.
Thus, **the answer is**: **No**, the new house after the transformation is not congruent to the original house.