The point Q(- 5, - 9) is rotated -270about the origin. Select the location of Q'
To rotate a point about the origin counterclockwise, we can use the following formula:
(x', y') = (x * cos(theta) - y * sin(theta), x * sin(theta) + y * cos(theta))
In this case, the angle of rotation is -270 degrees. Converting this to radians, we have theta = -270 * pi / 180 = -3pi / 2.
Let's substitute the coordinates of Q(-5, -9) into the formula:
(x', y') = (-5 * cos(-3pi / 2) - (-9) * sin(-3pi / 2), -5 * sin(-3pi / 2) + (-9) * cos(-3pi / 2))
Now, let's calculate:
(x', y') = (-5 * 0 - (-9) * (-1), -5 * (-1) + (-9) * 0)
= (0 + 9, 5 + 0)
= (9, 5)
Therefore, after rotating Q(-5, -9) counterclockwise by -270 degrees about the origin, the new location Q' is (9, 5).
To rotate a point about the origin, we use the following rotation formula:
For a point (x, y) rotated θ degrees counterclockwise about the origin, the new point (x', y') is given by:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, the point Q(-5, -9) is rotated -270 degrees about the origin. Let's substitute the values into the formula:
x' = (-5) * cos(-270) - (-9) * sin(-270)
y' = (-5) * sin(-270) + (-9) * cos(-270)
To evaluate these trigonometric values, remember that cos(-θ) = cos(θ) and sin(-θ) = -sin(θ).
x' = (-5) * cos(270) - (-9) * (-sin(270))
= (-5) * 0 - (-9) * (-1)
= 0 - 9
= -9
y' = (-5) * (-sin(270)) + (-9) * cos(270)
= (-5) * 1 + (-9) * 0
= -5 + 0
= -5
Therefore, the new location of Q' after rotating Q(-5, -9) -270 degrees about the origin is Q'(-9, -5).
To find the location of Q' after rotating point Q(-5, -9) about the origin by -270 degrees, we need to apply the rotation transformation.
Step 1: Find the new coordinates (x', y') using the following formulas:
x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)
where (x, y) are the original coordinates of point Q and theta is the angle of rotation (-270 degrees in this case).
Step 2: Substitute the values into the formulas:
x' = -5 * cos(-270) - (-9) * sin(-270)
y' = -5 * sin(-270) + (-9) * cos(-270)
Step 3: Simplify the equations using the trigonometric values:
x' = -5 * cos(90) - (-9) * sin(90)
y' = -5 * sin(90) + (-9) * cos(90)
Step 4: Calculate the trigonometric values:
cos(90) = 0
sin(90) = 1
cos(-90) = 0
sin(-90) = -1
Step 5: Substitute the trigonometric values:
x' = -5 * 0 - (-9) * (-1)
y' = -5 * 1 + (-9) * 0
Step 6: Simplify the equations:
x' = 0 + 9
y' = -5 + 0
Step 7: Final result:
x' = 9
y' = -5
Therefore, the new location Q' is (9, -5).