A line segment with endpoints at #(5 , -9 )# and #(2, -7 )# is rotated clockwise by #pi/2#. What are the new endpoints of the line segment?

Question

A line segment with endpoints at #(5 , -9   )# and #(2, -7    )# is rotated clockwise by #pi/2#.  What are the new endpoints of the line segment?

Maverick Smith · Accepted Answer

The endpoints of the new line segment are #(9,5)# and #(7,2)#.

Rotate each point #pi/2# radians, so I'll start with #(2,-7)# because it's closer to the origin. #(2,-7)# becomes #(7,2)# #(5,-9)# becomes #(9,5)#

Violet Aldrich · Answer

undefined To find the new endpoints of the line segment after rotating it clockwise by π/2, we can use the following transformation:

For a point (x, y) rotated clockwise by an angle θ about the origin, the new coordinates (x', y') are given by:
x' = x * cos(θ) + y * sin(θ)
y' = -x * sin(θ) + y * cos(θ)

Given the endpoints of the line segment as (5, -9) and (2, -7), we'll rotate each point individually by π/2.

For the first endpoint (5, -9):
x' = 5 * cos(π/2) + (-9) * sin(π/2) = 5 * 0 + (-9) * (-1) = 0 + 9 = 9
y' = -5 * sin(π/2) + (-9) * cos(π/2) = -5 * 1 + (-9) * 0 = -5 + 0 = -5
So, the new endpoint is (9, -5).

For the second endpoint (2, -7):
x' = 2 * cos(π/2) + (-7) * sin(π/2) = 2 * 0 + (-7) * (-1) = 0 + 7 = 7
y' = -2 * sin(π/2) + (-7) * cos(π/2) = -2 * 1 + (-7) * 0 = -2 + 0 = -2
So, the new endpoint is (7, -2).

Therefore, after rotating the line segment clockwise by π/2, the new endpoints are (9, -5) and (7, -2).