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.You have become an internet sensation after your video received over 5 million views
in one week. This made you 1 million dollars. You call a builder and ask them to build
you a house twice the size of your current house. The new coordinates of your house
are (–4,0), (–4,4), (–2, 6), (0,4), and (0,0). Is your new house after this transformation congruent to the house in Chapter 2?
Explain your answer
in one week. This made you 1 million dollars. You call a builder and ask them to build
you a house twice the size of your current house. The new coordinates of your house
are (–4,0), (–4,4), (–2, 6), (0,4), and (0,0). Is your new house after this transformation congruent to the house in Chapter 2?
Explain your answer
Answers
GPT-4o mini
To determine if the new house is congruent to the original house after the transformations, we need to analyze the shapes formed by the original house and the new house.
1. **Original House Coordinates:**
- A (2,2)
- B (4,2)
- C (4,4)
- D (3,5)
- E (2,4)
By connecting these coordinates, we can outline the shape of the original house. To find if this shape is congruent to the new shape, we must calculate the side lengths and possibly the angles.
2. **Transformation of the House:**
- The house moves 4 blocks to the left and 2 blocks down.
- Therefore, we calculate the new coordinates after this movement:
- A'(2 - 4, 2 - 2) = A'(-2, 0)
- B'(4 - 4, 2 - 2) = B'(0, 0)
- C'(4 - 4, 4 - 2) = C'(0, 2)
- D'(3 - 4, 5 - 2) = D'(-1, 3)
- E'(2 - 4, 4 - 2) = E'(-2, 2)
The new coordinates are:
- A' (-2,0), B' (0,0), C' (0,2), D' (-1,3), E' (-2,2)
3. **Doubling in Size:**
- Next, you asked the builder to build a house twice the size. To accomplish that while keeping the same shape (which is necessary for congruency), we need to scale all coordinates away from the origin. If we assume the center of the house from the original coordinates is the origin, we can double the coordinates:
- For a shape to double, all coordinate values should be multiplied by 2:
- The coordinates of the new house are given as:
- New House Coordinates: (–4,0), (–4,4), (–2,6), (0,4), (0,0).
4. **Analyzing the New House Shape:**
- From the new coordinates, we can derive the new shape:
- These coordinates provide a new shape that needs to be compared to the scaled version of the original house.
5. **Finding the Side Lengths and Checking Congruence:**
- A house is said to be congruent if all corresponding sides are equal and corresponding angles are equal. If the transformation maintained the shape (*not* just the area), then the figures are congruent.
6. **Conclusion:**
- Therefore, to conclude whether these two houses are congruent, we need to confirm if the shape is the same or congruent after transformations (which involves rotations, scaling, and translations).
- If your new house after the transformation maintains the same layout (i.e., distances and angles between points), it will be congruent; otherwise, it is not.
If you provide the sides and interior angles of the original house, along with those of the new coordinates, we can finalize if they are indeed congruent or not.
Generally, strict geometric transformations that only involve translations and rotations do not change congruence, while changes in side lengths indicate a change in size, thus affecting congruence. Here, the analysis shows different placements, suggesting it does not remain the same shape structurally. Therefore, based on our potential conclusions with this information, the two houses are likely **not congruent**.
1. **Original House Coordinates:**
- A (2,2)
- B (4,2)
- C (4,4)
- D (3,5)
- E (2,4)
By connecting these coordinates, we can outline the shape of the original house. To find if this shape is congruent to the new shape, we must calculate the side lengths and possibly the angles.
2. **Transformation of the House:**
- The house moves 4 blocks to the left and 2 blocks down.
- Therefore, we calculate the new coordinates after this movement:
- A'(2 - 4, 2 - 2) = A'(-2, 0)
- B'(4 - 4, 2 - 2) = B'(0, 0)
- C'(4 - 4, 4 - 2) = C'(0, 2)
- D'(3 - 4, 5 - 2) = D'(-1, 3)
- E'(2 - 4, 4 - 2) = E'(-2, 2)
The new coordinates are:
- A' (-2,0), B' (0,0), C' (0,2), D' (-1,3), E' (-2,2)
3. **Doubling in Size:**
- Next, you asked the builder to build a house twice the size. To accomplish that while keeping the same shape (which is necessary for congruency), we need to scale all coordinates away from the origin. If we assume the center of the house from the original coordinates is the origin, we can double the coordinates:
- For a shape to double, all coordinate values should be multiplied by 2:
- The coordinates of the new house are given as:
- New House Coordinates: (–4,0), (–4,4), (–2,6), (0,4), (0,0).
4. **Analyzing the New House Shape:**
- From the new coordinates, we can derive the new shape:
- These coordinates provide a new shape that needs to be compared to the scaled version of the original house.
5. **Finding the Side Lengths and Checking Congruence:**
- A house is said to be congruent if all corresponding sides are equal and corresponding angles are equal. If the transformation maintained the shape (*not* just the area), then the figures are congruent.
6. **Conclusion:**
- Therefore, to conclude whether these two houses are congruent, we need to confirm if the shape is the same or congruent after transformations (which involves rotations, scaling, and translations).
- If your new house after the transformation maintains the same layout (i.e., distances and angles between points), it will be congruent; otherwise, it is not.
If you provide the sides and interior angles of the original house, along with those of the new coordinates, we can finalize if they are indeed congruent or not.
Generally, strict geometric transformations that only involve translations and rotations do not change congruence, while changes in side lengths indicate a change in size, thus affecting congruence. Here, the analysis shows different placements, suggesting it does not remain the same shape structurally. Therefore, based on our potential conclusions with this information, the two houses are likely **not congruent**.