Given a point (x, y) on a geometric figure, what is the new point when the figure is rotated counterclockwise about the origin 180 degrees?(1 point)
The new point can be found by rotating the original point counterclockwise by 180 degrees about the origin.
To do this, we can use the following formulas for rotating a point (x, y) counterclockwise by angle θ about the origin:
New x-coordinate = x*cos(θ) - y*sin(θ)
New y-coordinate = x*sin(θ) + y*cos(θ)
In this case, since we want to rotate the point by 180 degrees, we substitute θ = π.
New x-coordinate = x*cos(π) - y*sin(π)
= -x - 0
= -x
New y-coordinate = x*sin(π) + y*cos(π)
= 0 - y
= -y
Therefore, the new point when the figure is rotated counterclockwise by 180 degrees is (-x, -y).
Given a point (1, 2) on a geometric figure, what is the new point when the figure is rotated clockwise about the origin 180 degrees?(1 point)
To find the new point when a figure is rotated clockwise about the origin by 180 degrees, we can use a similar approach as before.
Using the formulas for rotating a point (x, y) clockwise by angle θ about the origin:
New x-coordinate = x*cos(θ) + y*sin(θ)
New y-coordinate = -x*sin(θ) + y*cos(θ)
In this case, we want to rotate the point by 180 degrees, so we substitute θ = -π.
New x-coordinate = x*cos(-π) + y*sin(-π)
= x*(-1) + y*0
= -x
New y-coordinate = -x*sin(-π) + y*cos(-π)
= x*0 + y*(-1)
= -y
Therefore, the new point when the figure is rotated clockwise by 180 degrees is (-x, -y), which in this case would be (-1, -2).
Given a point (−6, 4) in a line segment, what is the new point when the line segment is rotated counterclockwise about the origin 180 degrees?(1 point)
To find the new point when a line segment is rotated counterclockwise about the origin by 180 degrees, we can use the same approach as before.
Using the formulas for rotating a point (x, y) counterclockwise by angle θ about the origin:
New x-coordinate = x*cos(θ) - y*sin(θ)
New y-coordinate = x*sin(θ) + y*cos(θ)
In this case, we want to rotate the point by 180 degrees, so we substitute θ = π.
New x-coordinate = (-6)*cos(π) - 4*sin(π)
= (-6)*(-1) - 4*0
= 6
New y-coordinate = (-6)*sin(π) + 4*cos(π)
= (-6)*0 + 4*(-1)
= -4
Therefore, the new point when the line segment is rotated counterclockwise by 180 degrees is (6, -4).
Use the image to answer the question.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 8 to 8 in increments of 1. Four triangles are plotted on the graph. The four triangles plotted are triangle A B C, triangle A prime B prime C prime, triangle A double prime B double prime C double prime, and triangle A triple prime B triple prime C triple prime. The triangle A B C has its vertices marked with closed points at A left parenthesis 2 comma 6 right parenthesis, B left parenthesis 6 comma 4 right parenthesis, and C left parenthesis 2 comma 2 right parenthesis. The triangle A prime B prime C prime has its vertices marked with closed points at A prime left parenthesis negative 6 comma 2 right parenthesis, B prime left parenthesis negative 4 comma 6 right parenthesis, and C prime left parenthesis negative 2 comma 2 right parenthesis. The triangle A double prime B double prime C double prime has its vertices marked with closed points at A double prime left parenthesis negative 2 comma negative 6 right parenthesis, B double prime left parenthesis negative 6 comma negative 4 right parenthesis, and C double prime left parenthesis negative 2 comma negative 2 right parenthesis. The triangle A triple prime B triple prime C triple prime has its vertices marked with closed points at A triple prime left parenthesis 6 comma negative 2 right parenthesis, B triple prime left parenthesis 4 comma negative 6 right parenthesis, and C triple prime left parenthesis 2 comma negative 2 right parenthesis.
Given point A(2, 6) on the triangle, where would the new point on the triangle be after a rotation of 90 degrees clockwise?
(1 point)
()
To find the new point after a rotation of 90 degrees clockwise, we can use the transformation matrix for a clockwise rotation by 90 degrees about the origin:
```
[ cos(90°) -sin(90°) ]
[ sin(90°) cos(90°) ]
```
Substituting the values from point A (2, 6):
```
[ 0 -1 ]
[ 1 0 ]
```
To apply this transformation matrix to point A (2, 6):
```
New x-coordinate = 0*2 - (-1)*6
= 0 + 6
= 6
New y-coordinate = 1*2 + 0*6
= 2 + 0
= 2
```
Therefore, the new point on the triangle after a rotation of 90 degrees clockwise from point A (2, 6) would be (6, 2).
Use the image to answer the question.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 8 to 8 in increments of 1. A triangle A B C is plotted on the graph. The triangle A B C has its vertices marked with closed points at A left parenthesis 2 comma 6 right parenthesis, B left parenthesis 6 comma 4 right parenthesis, and C left parenthesis 2 comma 2 right parenthesis.
Given point A(2, 6) on the triangle, where would the new point on the triangle be after a rotation of 90 degrees counterclockwise?
(1 point)
()
To find the new point after a rotation of 90 degrees counterclockwise, we can use the transformation matrix for a counterclockwise rotation by 90 degrees about the origin:
```
[ cos(90°) -sin(90°) ]
[ sin(90°) cos(90°) ]
```
Substituting the values from point A (2, 6):
```
[ 0 -1 ]
[ 1 0 ]
```
To apply this transformation matrix to point A (2, 6):
```
New x-coordinate = 0*2 - (-1)*6
= 0 + 6
= 6
New y-coordinate = 1*2 + 0*6
= 2 + 0
= 2
```
Therefore, the new point on the triangle after a rotation of 90 degrees counterclockwise from point A (2, 6) would be (6, 2).