A tornado hit the area. It picked up your house and rotated it 180 degrees

Question

counterclockwise.
Question 9
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.
Question 10
Is the new location of your house after this transformation congruent to the
neighbor’s house in Chapter 4? Explain your answer.

Answer 1

To answer your questions, we will first perform a 180-degree counterclockwise rotation of a point around the origin, and then we'll analyze the congruence of the transformed house location compared to the neighbor's house.

Question 9: Rotating points 180 degrees counterclockwise

1. Understanding the Rotation:

   • The formula for rotating a point \((x, y)\) 180 degrees counterclockwise about the origin is: \[ (x, y) \rightarrow (-x, -y) \]

2. Original Points:

   • Let's say the original points of your house are given as: \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\), ...
   • For example, if your points were \((2, 3)\), \((4, 5)\), and \((6, 1)\).

3. Rotating the Points:

   • Applying the rotation formula:
     • \((2, 3) \rightarrow (-2, -3)\)
     • \((4, 5) \rightarrow (-4, -5)\)
     • \((6, 1) \rightarrow (-6, -1)\)

4. Color Choice:

   • You can choose any color for plotting. For this example, let’s say blue.

5. New Points of Your House After Rotation:

   • The new points after a 180-degree counterclockwise rotation would be:
     • \((-2, -3)\)
     • \((-4, -5)\)
     • \((-6, -1)\)

Question 10: Congruence of the new location of your house

1. Definition of Congruence:

   • Two shapes are congruent if one can be transformed into the other using rotations, translations, reflections, or a combination of these, without changing its size or shape.

2. Comparison to Neighbor’s House (Assuming Neighbor's Coordinates):

   • If the neighbor's house, for instance, is represented by the points \((-2, -3)\), \((-4, -5)\), and \((-6, -1)\) as well, then:
   • Your new house points are identical to your neighbor's house points.
   • Therefore, your house is congruent to your neighbor's house since they occupy the same points in space.

3. Conclusion:

   • Yes, the new location of your house after this transformation is congruent to the neighbor's house if the coordinates match. If not, it would depend on the relative positions of both sets of points.

Overall, after completing these calculations, create your graph using blue for the new points of your rotated house, and check the coordinates against your neighbor's house to determine congruence.