Question

The point (5,-2) is rotated 270° counterclockwise around the origin and then dilated by a scale factor of 2 at the origin. What are the coordinates of the resulting image?

a) (-4,-10)
b) (10,-4)
c) (-2,-5)
d) (2,5)

Answer 1

The coordinates of the resulting image after rotating the point (5,-2) 270° counterclockwise around the origin and then dilating by a scale factor of 2 at the origin are (-4, -10).

Step-by-step explanation:

To rotate a point counterclockwise around the origin, we can use the rotation matrix:

x' = x*cos(theta) - y*sin(theta)
y' = x*sin(theta) + y*cos(theta)

For a rotation of 270° counterclockwise, we have theta = 270° = 3*pi/2 radians:

x' = 5*cos(3*pi/2) - (-2)*sin(3*pi/2) = 5*0 - (-2)*(-1) = 0 - 2 = -2
y' = 5*sin(3*pi/2) + (-2)*cos(3*pi/2) = 5*(-1) + (-2)*0 = -5

To dilate a point by a scale factor of 2 at the origin, we can multiply the coordinates by 2.

x'' = -2 * 2 = -4
y'' = -5 * 2 = -10

The resulting coordinates are (-4, -10).