What are the coordinate points of A′ if A (5, 6) undergoes a rotation of:
90° clockwise:
270° clockwise:
To find the coordinate points of A' after a rotation of 90° clockwise and 270° clockwise, we can use the rotation matrix formula:
For a rotation of 90° clockwise:
x' = x cos(theta) - y sin(theta)
y' = x sin(theta) + y cos(theta)
Using A(5, 6), we have:
x = 5
y = 6
theta = 90°
Plugging these values into the rotation matrix formula:
x' = 5 cos(90°) - 6 sin(90°)
= 5 (0) - 6 (1)
= -6
y' = 5 sin(90°) + 6 cos(90°)
= 5 (1) + 6 (0)
= 5
Therefore, the coordinate points of A' after a rotation of 90° clockwise is A'(-6, 5).
For a rotation of 270° clockwise:
x' = x cos(theta) - y sin(theta)
y' = x sin(theta) + y cos(theta)
Using A(5, 6), we have:
x = 5
y = 6
theta = 270°
Plugging these values into the rotation matrix formula:
x' = 5 cos(270°) - 6 sin(270°)
= 5 (0) - 6 (-1)
= 6
y' = 5 sin(270°) + 6 cos(270°)
= 5 (-1) + 6 (0)
= -5
Therefore, the coordinate points of A' after a rotation of 270° clockwise is A'(6, -5).
What are the coordinate points of A′ if A (5, 6) undergoes a rotation of:
180° clockwise:
180° counterclockwise:
To find the coordinate points of A' after a rotation of 180° clockwise and 180° counterclockwise, we again use the rotation matrix formula.
For a rotation of 180° clockwise:
x' = x cos(theta) - y sin(theta)
y' = x sin(theta) + y cos(theta)
Using A(5, 6), we have:
x = 5
y = 6
theta = 180°
Plugging these values into the rotation matrix formula:
x' = 5 cos(180°) - 6 sin(180°)
= 5 (-1) - 6 (0)
= -5
y' = 5 sin(180°) + 6 cos(180°)
= 5 (0) + 6 (-1)
= -6
Therefore, the coordinate points of A' after a rotation of 180° clockwise is A'(-5, -6).
For a rotation of 180° counterclockwise, we can simply take the negative of the coordinate points after a 180° clockwise rotation. Therefore, the coordinate points of A' after a rotation of 180° counterclockwise are A'(5, 6).
180° counterclockwise:
To find the coordinate points of A' after a rotation of 180° counterclockwise, we use the same rotation matrix formula:
x' = x cos(theta) - y sin(theta)
y' = x sin(theta) + y cos(theta)
Using A(5, 6), we have:
x = 5
y = 6
theta = 180°
Plugging these values into the rotation matrix formula:
x' = 5 cos(180°) - 6 sin(180°)
= 5 (-1) - 6 (0)
= -5
y' = 5 sin(180°) + 6 cos(180°)
= 5 (0) + 6 (1)
= 6
Therefore, the coordinate points of A' after a rotation of 180° counterclockwise is A'(-5, 6).