Point A is on figure B and is reflected across the x-axis. if point has coordinates of (3, 5), what is the distance between the original point A and the reflected point A?
When a point is reflected across the x-axis, the y-coordinate is negated.
If point A has coordinates (3, 5), the reflection across the x-axis would have coordinates (3, -5).
The distance between two points can be calculated using the distance formula:
√[(x2 - x1)² + (y2 - y1)²]
In this case, the distance formula becomes:
√[(3 - 3)² + (-5 - 5)²]
√[(0)² + (-10)²]
√[0 + 100]
√100
10
Therefore, the distance between the original point A and the reflected point A is 10 units.
To rotate a point 180 degrees clockwise on a line segment, we can use the following rotation formula:
x' = x * cos(angle) + y * sin(angle)
y' = -x * sin(angle) + y * cos(angle)
Given the point (9, 5) and rotating it 180 degrees clockwise, we can substitute the values into the rotation formula:
x' = (9) * cos(180) + (5) * sin(180)
y' = -(9) * sin(180) + (5) * cos(180)
Now, let's calculate the cosine and sine values for 180 degrees:
cos(180) = -1
sin(180) = 0
Substituting these values into the rotation formula:
x' = (9) * (-1) + (5) * 0
y' = -(9) * 0 + (5) * (-1)
Simplifying:
x' = -9 + 0
y' = 0 - 5
x' = -9
y' = -5
Therefore, the new point after rotating (9, 5) 180 degrees clockwise is (-9, -5).
If a point (x, y) in quadrant I is rotated 180 degrees clockwise, it will end up in quadrant III.
Given point A (2, 4) in quadrant I, the new point after rotating 180 degrees clockwise will have the same x-coordinate but a negated y-coordinate.
So, the new point will have coordinates (-2, -4) and it will be located in quadrant III.
rotate the point (-3, -5) on a geometric figure 270 degrees counterclockwise. what Is the new point?
To rotate a point counterclockwise on a geometric figure, we can use the following rotation formula:
x' = x * cos(angle) - y * sin(angle)
y' = x * sin(angle) + y * cos(angle)
Given the point (-3, -5) and rotating it 270 degrees counterclockwise, we can substitute the values into the rotation formula:
x' = (-3) * cos(270) - (-5) * sin(270)
y' = (-3) * sin(270) + (-5) * cos(270)
Now, let's calculate the cosine and sine values for 270 degrees:
cos(270) = 0
sin(270) = -1
Substituting these values into the rotation formula:
x' = (-3) * 0 - (-5) * (-1)
y' = (-3) * (-1) + (-5) * 0
Simplifying:
x' = 0 - 5
y' = 3 + 0
x' = -5
y' = -3
Therefore, the new point after rotating (-3, -5) 270 degrees counterclockwise is (-5, -3).