To break down the problem step-by-step, we will first address each chapter and the transformations described. We will clarify the points involved, the nature of the transformations, and congruence.
Chapter 1: Original House
The original points of your house are:
- A(2, 3)
- B(4, 3)
- C(4, 5)
- D(3, 6)
- E(2, 5)
Chapter 2: Flood Transformation
The flood moves your house 4 blocks to the left and 2 blocks down.
This means we are applying the transformation:
- New x-coordinate = Old x-coordinate - 4
- New y-coordinate = Old y-coordinate - 2
New Coordinates
- A(2-4, 3-2) = A'(-2, 1)
- B(4-4, 3-2) = B'(0, 1)
- C(4-4, 5-2) = C'(0, 3)
- D(3-4, 6-2) = D'(-1, 4)
- E(2-4, 5-2) = E'(-2, 3)
The new points are:
- A'(-2, 1)
- B'(0, 1)
- C'(0, 3)
- D'(-1, 4)
- E'(-2, 3)
Description of Transformation: The transformation that occurred is a translation, where every point was moved 4 units left (decreasing x) and 2 units down (decreasing y).
Congruence Check: Yes, the new house after this translation is congruent to the original house. A translation does not alter the shape or size of the figure.
Chapter 3: Neighbor’s House
Your neighbor builds the exact same house as yours, with the points of your house after the flood being reflected over the x-axis.
Reflection Transformation over the x-axis involves changing the y-coordinate from y to -y (the x-coordinates remain the same).
New Neighbor's Coordinates:
- A'(-2, 1) → A''(-2, -1)
- B'(0, 1) → B''(0, -1)
- C'(0, 3) → C''(0, -3)
- D'(-1, 4) → D''(-1, -4)
- E'(-2, 3) → E''(-2, -3)
The points of the neighbor’s house are:
- A''(-2, -1)
- B''(0, -1)
- C''(0, -3)
- D''(-1, -4)
- E''(-2, -3)
Color Used: Let’s say the color used for the graph is blue.
Congruence Check: Yes, the neighbor's house is congruent to your house. A reflection does not change the size or shape of the house but flips it over the x-axis.
Chapter 4: Tornado Transformation
The tornado rotates your house (from Chapter 2) 180 degrees counterclockwise about the origin.
New Coordinates after 180-degree rotation:
To find the new coordinates after a 180-degree rotation, we can use the transformation:
- New x-coordinate = -Old x-coordinate
- New y-coordinate = -Old y-coordinate
New Coordinates:
- A'(-2, 1) → A'''(2, -1)
- B'(0, 1) → B'''(0, -1)
- C'(0, 3) → C'''(0, -3)
- D'(-1, 4) → D'''(1, -4)
- E'(-2, 3) → E'''(2, -3)
The new points of your house after the tornado are:
- A'''(2, -1)
- B'''(0, -1)
- C'''(0, -3)
- D'''(1, -4)
- E'''(2, -3)
Question 8: Congruence Check with Neighbor’s House
To check for congruence with your neighbor's house (which is in blue):
- Your new points are:
- A'''(2, -1)
- B'''(0, -1)
- C'''(0, -3)
- D'''(1, -4)
- E'''(2, -3)
The neighbor’s points are:
- A''(-2, -1)
- B''(0, -1)
- C''(0, -3)
- D''(-1, -4)
- E''(-2, -3)
Final Congruence Check: No, your house after the tornado transformation is not congruent to the neighbor's house. The transformation involved a rotation, which results in a different orientation of the shape, altering its position in relation to the x-axis, even though the shape and size remain the same.
In summary, throughout these transformations, the shape of your house remained congruent to its original and the neighbor's, while the placements changed due to translation, reflection, and rotation.