To rotate a point counterclockwise by 180 degrees about the origin, we use the following transformation matrix:
\[ \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix} \]
In this case, since we want to rotate by 180 degrees, we have:
\[ \begin{bmatrix} cos(180^\circ) & -sin(180^\circ) \\ sin(180^\circ) & cos(180^\circ) \end{bmatrix} \]
Simplifying this, we get:
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \]
Now, let's apply this transformation matrix to each point:
For (-8, 8):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -8 \\ 8 \end{bmatrix} = \begin{bmatrix} 8 \\ -8 \end{bmatrix} \]
For (-2, 6):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -2 \\ 6 \end{bmatrix} = \begin{bmatrix} 2 \\ -6 \end{bmatrix} \]
For (-4, 12):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -4 \\ 12 \end{bmatrix} = \begin{bmatrix} 4 \\ -12 \end{bmatrix} \]
For (0, 8):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 0 \\ 8 \end{bmatrix} = \begin{bmatrix} 0 \\ -8 \end{bmatrix} \]
For (0, 0):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]
So, the points of the new house after the counterclockwise rotation of 180 degrees about the origin are:
(-8, 8) becomes (8, -8)
(-2, 6) becomes (2, -6)
(-4, 12) becomes (4, -12)
(0, 8) becomes (0, -8)
(0, 0) remains the same at (0, 0)
Question 9
Plot and state the points of your new house from Chapter (-8, 8), (-2, 6), (-4,12), (0, 8), and (0, 0) 3 after a counterclockwise rotation of 180 degrees about the origin.
5 answers
Question 9
Plot and state the points of your new house from Chapter 3 (-4, 0), (-4, 4), (-2, 6), 10,4), and (0, 0).
after a counterclockwise rotation of 180 degrees about the origin.
Plot and state the points of your new house from Chapter 3 (-4, 0), (-4, 4), (-2, 6), 10,4), and (0, 0).
after a counterclockwise rotation of 180 degrees about the origin.
To rotate a point counterclockwise by 180 degrees about the origin, we use the following transformation matrix:
\[ \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix} \]
In this case, since we want to rotate by 180 degrees, we have:
\[ \begin{bmatrix} cos(180^\circ) & -sin(180^\circ) \\ sin(180^\circ) & cos(180^\circ) \end{bmatrix} \]
Simplifying this, we get:
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \]
Now, let's apply this transformation matrix to each point:
For (-4, 0):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -4 \\ 0 \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \end{bmatrix} \]
For (-4, 4):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -4 \\ 4 \end{bmatrix} = \begin{bmatrix} 4 \\ -4 \end{bmatrix} \]
For (-2, 6):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -2 \\ 6 \end{bmatrix} = \begin{bmatrix} 2 \\ -6 \end{bmatrix} \]
For (10, 4):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 10 \\ 4 \end{bmatrix} = \begin{bmatrix} -10 \\ -4 \end{bmatrix} \]
For (0, 0):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]
So, the points of the new house after the counterclockwise rotation of 180 degrees about the origin are:
(-4, 0) becomes (4, 0)
(-4, 4) becomes (4, -4)
(-2, 6) becomes (2, -6)
(10, 4) becomes (-10, -4)
(0, 0) remains the same at (0, 0)
\[ \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{bmatrix} \]
In this case, since we want to rotate by 180 degrees, we have:
\[ \begin{bmatrix} cos(180^\circ) & -sin(180^\circ) \\ sin(180^\circ) & cos(180^\circ) \end{bmatrix} \]
Simplifying this, we get:
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \]
Now, let's apply this transformation matrix to each point:
For (-4, 0):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -4 \\ 0 \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \end{bmatrix} \]
For (-4, 4):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -4 \\ 4 \end{bmatrix} = \begin{bmatrix} 4 \\ -4 \end{bmatrix} \]
For (-2, 6):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -2 \\ 6 \end{bmatrix} = \begin{bmatrix} 2 \\ -6 \end{bmatrix} \]
For (10, 4):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 10 \\ 4 \end{bmatrix} = \begin{bmatrix} -10 \\ -4 \end{bmatrix} \]
For (0, 0):
\[ \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]
So, the points of the new house after the counterclockwise rotation of 180 degrees about the origin are:
(-4, 0) becomes (4, 0)
(-4, 4) becomes (4, -4)
(-2, 6) becomes (2, -6)
(10, 4) becomes (-10, -4)
(0, 0) remains the same at (0, 0)
Question 10
Is the new location of your house after this transformation congruent to the neighbor's house in Chapter 4 (8, 8), (2, 6), (4, 12), (0, 8), and (0, 0), ? Explain your answer.
Is the new location of your house after this transformation congruent to the neighbor's house in Chapter 4 (8, 8), (2, 6), (4, 12), (0, 8), and (0, 0), ? Explain your answer.
No, the new location of my house after the rotation of 180 degrees about the origin is not congruent to the neighbor's house in Chapter 4.
Congruent shapes are identical in size and shape, but they may be in different positions or orientations. In this case, the points of my new house after the rotation are the mirror image of the neighbor's house in Chapter 4. The coordinates are reflected across the x-axis and y-axis, resulting in a different position and orientation.
For example, while both houses have the point (8, 8), in my new house, this point becomes (-8, -8). Similarly, the other points also have different coordinate values.
Therefore, the new location of my house after the transformation is not congruent to the neighbor's house in Chapter 4.
Congruent shapes are identical in size and shape, but they may be in different positions or orientations. In this case, the points of my new house after the rotation are the mirror image of the neighbor's house in Chapter 4. The coordinates are reflected across the x-axis and y-axis, resulting in a different position and orientation.
For example, while both houses have the point (8, 8), in my new house, this point becomes (-8, -8). Similarly, the other points also have different coordinate values.
Therefore, the new location of my house after the transformation is not congruent to the neighbor's house in Chapter 4.
1 answer