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 (x, y) on a geometric figure, what is the new point when the figure is rotated counterclockwise about the origin 180 degrees?(1 point)
9 answers
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).
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).
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)
()
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).
```
[ 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)
()
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).
```
[ 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).
1 answer