Graph XY with endpoints X(5,−2)and Y(3,−3)and its image after a reflection in the x-axis and then a rotation of 270 degrees about the origin.
What are the following coordinates for XY after a reflection in the x-axis and then a rotation of 270 degrees about the orgin?
Please show all the work on how you got your answer.
To find the coordinates of XY after a reflection in the x-axis and then a rotation of 270 degrees about the origin, we can follow these steps:
Step 1: Reflection in the x-axis
To reflect a point (x, y) in the x-axis, we keep the x-coordinate the same and change the sign of the y-coordinate.
X(5,-2) with reflection: (5, 2)
Y(3,-3) with reflection: (3, 3)
Step 2: Rotation of 270 degrees about the origin
To rotate a point (x, y) about the origin by 270 degrees counterclockwise, we can use the following rotation formula:
x' = x * cos(270) - y * sin(270)
y' = x * sin(270) + y * cos(270)
Using these formulas, we can calculate the new coordinates after the rotation.
For point X'(5, 2):
x' = 5 * cos(270) - 2 * sin(270)
= 5 * 0 - 2 * (-1)
= 0 + 2
= 2
y' = 5 * sin(270) + 2 * cos(270)
= 5 * (-1) + 2 * 0
= -5 + 0
= -5
Therefore, point X(5, 2) after a rotation of 270 degrees about the origin is X'(2, -5).
For point Y'(3, 3):
x' = 3 * cos(270) - 3 * sin(270)
= 3 * 0 - 3 * (-1)
= 0 + 3
= 3
y' = 3 * sin(270) + 3 * cos(270)
= 3 * (-1) + 3 * 0
= -3 + 0
= -3
Therefore, point Y(3, 3) after a rotation of 270 degrees about the origin is Y'(3, -3).
The coordinates of the points XY after a reflection in the x-axis and then a rotation of 270 degrees about the origin are:
X'(2, -5)
Y'(3, -3)
To find the coordinates for XY after a reflection in the x-axis and then a rotation of 270 degrees about the origin, we will follow these steps:
1. Reflection in the x-axis: To reflect a point (x, y) in the x-axis, we need to change the sign of the y-coordinate. In other words, if the original point is (x, y), the reflected point will be (x, -y).
Let's find the reflected point from the original points X(5,−2):
The reflection of X in the x-axis will be X' = (5, -(-2)) = (5, 2).
Similarly, let's find the reflected point from the original points Y(3,−3):
The reflection of Y in the x-axis will be Y' = (3, -(-3)) = (3, 3).
2. Rotation of 270 degrees about the origin: To rotate a point (x, y) by 270 degrees about the origin, we need to apply the following transformation:
x' = x*cos(270) - y*sin(270)
y' = x*sin(270) + y*cos(270)
Note: Since cosine of 270 degrees is 0 and sine of 270 degrees is -1, the formulas simplify to:
x' = y
y' = -x
Let's find the rotated points after reflection:
For X', X_rot = (2, 5) by substituting x=5 and y=2 into the formulas:
x' = 2
y' = -5
For Y', Y_rot = (3, -3) by substituting x=3 and y=3 into the formulas:
x' = -3
y' = -3
Therefore, the coordinates for XY after a reflection in the x-axis and then a rotation of 270 degrees about the origin are:
X_rot = (2, 5)
Y_rot = (-3, -3)
To summarize:
After reflecting X(5,−2) and Y(3,−3) in the x-axis and then rotating them 270 degrees about the origin, the transformed points are:
X_rot = (2, 5)
Y_rot = (-3, -3)