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move in across the street after they build the same exact house as you did in Chapter

3.
Question 7
With a different color, plot and state the points of your neighbor’s house, which is a
reflection of your house in Chapter 3 over the y-axis. Name the color you use on your
graph here.
Question 8
Is your new house after this transformation congruent to the house in Chapter 3?
Explain your answer. Plot and state the points of your new house from Chapter 3 after a counterclockwise
rotation of 180 degrees about the origin. Use a different color to plot the points.
Name the color you use on your graph here. Is the new location of your house after this transformation congruent to the
neighbor’s house in Chapter 4? Explain your answer.

1 answer

To answer Questions 7 and 8 thoroughly, let's break down the concepts of reflection, rotation, and congruence.

Question 7:

Reflection of your house (located at points in Chapter 3) over the y-axis means if you have a point \( (x, y) \), its reflection will be \( (-x, y) \). If your house had the following points (for example purposes):

  • A(1, 2)
  • B(3, 4)
  • C(5, 1)

The reflected points over the y-axis would be:

  • A'(-1, 2)
  • B'(-3, 4)
  • C'(-5, 1)

For the new house (your neighbor's house), let’s assign a color for this plot: Red.

Question 8:

To determine if your new house (after the 180-degree rotation) is congruent to your neighbor's house, we first need to rotate your points around the origin.

Rotation of 180 degrees about the origin means transforming \( (x, y) \) coordinates to \( (-x, -y) \).

Using the previous points from Chapter 3 (let’s take the same example):

  • A(1, 2) becomes A''(-1, -2)
  • B(3, 4) becomes B''(-3, -4)
  • C(5, 1) becomes C''(-5, -1)

For the rotated points, you can choose a different color to visualize this on your graph. Let’s use Blue for this transformation.

Now, you can check congruence:

  1. Two shapes are congruent if all their corresponding sides and angles are the same.
  2. Since your rotation by 180 degrees and your neighbor’s reflection over the y-axis did not change the shape's size or angles, both houses remain congruent.

Summary Answers:

  • Question 7: The points of the neighbor’s house, reflected over the y-axis, are plotted in Red.
  • Question 8: Your new house after a 180-degree rotation is represented by points in Blue. Both configurations (your house after the rotation and your neighbor's house) are congruent because they are transformations of each other (one is a reflection, and the other is a rotation) that preserve size and angles.