The point Q(−5,−9)
is rotated −270°
about the origin. Select the location of Q′
.
The point Q(-5, -9) rotated -270° about the origin will result in the point Q'(9, -5).
To rotate a point about the origin, we use the following formula:
Q'(x', y') = (x*cosθ - y*sinθ, x*sinθ + y*cosθ)
Where (x, y) are the original coordinates of the point, θ is the angle of rotation (in degrees), and (x', y') are the coordinates of the rotated point.
In this case, the original coordinates of point Q are (-5, -9), and we want to rotate it -270° about the origin.
Using the formula, we get:
Q'(x', y') = (-5*cos(-270°) - (-9)*sin(-270°), -5*sin(-270°) + (-9)*cos(-270°))
Now, let's calculate the rotated coordinates:
Q'(x', y') = (-5*cos(-270°) - (-9)*sin(-270°), -5*sin(-270°) + (-9)*cos(-270°))
= (-5*(-1) - (-9)*0, -5*0 + (-9)*(-1))
= (-5 + 0, 0 + 9)
= (-5, 9)
Therefore, the location of Q' is (-5, 9).
To find the coordinates of a point after rotating it about the origin, you can use the following steps:
Step 1: Determine the angle of rotation in degrees. In this case, the angle of rotation is -270°.
Step 2: Convert the given angle of rotation to radians. To convert from degrees to radians, you can use the formula: radians = (degrees * π) / 180. Let's calculate it:
radians = (-270 * π) / 180
= (-3π) / 2
Step 3: Apply the rotation transformation to the coordinates of the point Q(-5, -9). The rotation transformation is given by the formulas:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
Substituting the values, we have:
x' = (-5) * cos((-3π) / 2) - (-9) * sin((-3π) / 2)
y' = (-5) * sin((-3π) / 2) + (-9) * cos((-3π) / 2)
Now, let's calculate cosine and sine of (-3π) / 2:
cos((-3π) / 2) = 0
sin((-3π) / 2) = -1
Substituting these values:
x' = (-5) * 0 - (-9) * (-1) = 0 - 9 = -9
y' = (-5) * (-1) + (-9) * 0 = 5 + 0 = 5
Therefore, the location of Q' is Q' (-9, 5).