To rotate a point 90 degrees counterclockwise, we can use the rotation matrix:
\[\begin{bmatrix} \cos(90^\circ) & -\sin(90^\circ) \\ \sin(90^\circ) & \cos(90^\circ) \\ \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix}\]
Multiplying this matrix by the coordinates of the given point: (-7, 4)
\[\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} -7 \\ 4 \end{bmatrix}\]
This results in:
\[\begin{bmatrix} (-7) \cdot 0 + 4 \cdot (-1) \\ (-7) \cdot 1 + 4 \cdot 0 \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -4 \\ -7 \end{bmatrix}\]
Therefore, after rotating the figure 90 degrees counterclockwise, the new point is (-4, -7).
Given a point (-7,4) on a geometric figure, what is th new point after rotation the figure 90 degrees counterclockwise?
5 answers
Given a point (9, -3) on a geometric figure, what is the new point when the figure is rotated counterclockwise about the origin 180 degrees?
A. (9,3)
B. (-9,3)
C. (-9,-3)
D. (-3,9)
A. (9,3)
B. (-9,3)
C. (-9,-3)
D. (-3,9)
To rotate a point 180 degrees counterclockwise about the origin, we can use the rotation matrix:
\[\begin{bmatrix} \cos(180^\circ) & -\sin(180^\circ) \\ \sin(180^\circ) & \cos(180^\circ) \\ \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ \end{bmatrix}\]
Multiplying this matrix by the coordinates of the given point: (9,-3)
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \cdot \begin{bmatrix} 9 \\ -3 \end{bmatrix}\]
This results in:
\[\begin{bmatrix} (-1) \cdot 9 + 0 \cdot (-3) \\ 0 \cdot 9 + (-1) \cdot (-3) \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -9 \\ 3 \end{bmatrix}\]
Therefore, after rotating the figure 180 degrees counterclockwise about the origin, the new point is (-9, 3).
Therefore, the correct answer is B. (-9, 3).
\[\begin{bmatrix} \cos(180^\circ) & -\sin(180^\circ) \\ \sin(180^\circ) & \cos(180^\circ) \\ \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ \end{bmatrix}\]
Multiplying this matrix by the coordinates of the given point: (9,-3)
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \cdot \begin{bmatrix} 9 \\ -3 \end{bmatrix}\]
This results in:
\[\begin{bmatrix} (-1) \cdot 9 + 0 \cdot (-3) \\ 0 \cdot 9 + (-1) \cdot (-3) \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -9 \\ 3 \end{bmatrix}\]
Therefore, after rotating the figure 180 degrees counterclockwise about the origin, the new point is (-9, 3).
Therefore, the correct answer is B. (-9, 3).
Given point A(2,6) on the triangle, what are the new coordinates of the given point after the triangle is rotated 180 degrees counterclokwise?
To rotate a point 180 degrees counterclockwise, we can use the rotation matrix:
\[\begin{bmatrix} \cos(180^\circ) & -\sin(180^\circ) \\ \sin(180^\circ) & \cos(180^\circ) \\ \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ \end{bmatrix}\]
Multiplying this matrix by the coordinates of the given point: A(2,6)
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \cdot \begin{bmatrix} 2 \\ 6 \end{bmatrix}\]
This results in:
\[\begin{bmatrix} (-1) \cdot 2 + 0 \cdot 6 \\ 0 \cdot 2 + (-1) \cdot 6 \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -2 \\ -6 \end{bmatrix}\]
Therefore, after rotating the triangle 180 degrees counterclockwise, the new coordinates of point A are (-2, -6).
\[\begin{bmatrix} \cos(180^\circ) & -\sin(180^\circ) \\ \sin(180^\circ) & \cos(180^\circ) \\ \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ \end{bmatrix}\]
Multiplying this matrix by the coordinates of the given point: A(2,6)
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \cdot \begin{bmatrix} 2 \\ 6 \end{bmatrix}\]
This results in:
\[\begin{bmatrix} (-1) \cdot 2 + 0 \cdot 6 \\ 0 \cdot 2 + (-1) \cdot 6 \end{bmatrix}\]
which simplifies to:
\[\begin{bmatrix} -2 \\ -6 \end{bmatrix}\]
Therefore, after rotating the triangle 180 degrees counterclockwise, the new coordinates of point A are (-2, -6).
1 answer