If a segment has endpoints (0, 4) and (3, 4) , how long will the segment be after it is rotated 270° clockwise?(1 point)

Answer 1

To rotate a segment 270° clockwise, we can use the following formula:

(x', y') = (x * cosθ - y * sinθ, x * sinθ + y * cosθ)

In this case, the segment has endpoints (0, 4) and (3, 4). Let's rotate these points 270° clockwise:

For the first endpoint (0, 4):
x' = 0 * cos(270°) - 4 * sin(270°) = 0 - 4 * (-1) = 4
y' = 0 * sin(270°) + 4 * cos(270°) = 0 + 4 * 0 = 0

So the new coordinates for the first endpoint are (4, 0).

For the second endpoint (3, 4):
x' = 3 * cos(270°) - 4 * sin(270°) = 3 * 0 - 4 * (-1) = 4
y' = 3 * sin(270°) + 4 * cos(270°) = 3 * (-1) + 4 * 0 = -3

So the new coordinates for the second endpoint are (4, -3).

Now we can calculate the length of the segment with these new endpoints using the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((4 - 4)^2 + (-3 - 0)^2)
= sqrt(0 + 9)
= sqrt(9)
= 3

Therefore, the segment will be 3 units long after it is rotated 270° clockwise.