f a segment has endpoints (0, 4) and (3, 4) , how long will the segment be after it is rotated 270° clockwise?
To rotate a segment 270° clockwise, we can find the new coordinates of the endpoints after the rotation.
Let's first find the midpoint of the segment. The midpoint is the average of the x-coordinates and the average of the y-coordinates:
Midpoint: ((0 + 3) / 2, (4 + 4) / 2) = (1.5, 4)
Now, let's rotate the segment 270° clockwise. The rotation can be thought of as a translation of the entire coordinate plane 90° counterclockwise, followed by a reflection over the y-axis.
1. Translation of 90° counterclockwise: (x, y) → (-y, x)
Rotated endpoint 1: (4, -0) = (4, 0)
2. Reflection over the y-axis: (x, y) → (-x, y)
Final rotated endpoint 1: (-4, 0)
1. Translation of 90° counterclockwise: (x, y) → (-y, x)
Rotated endpoint 2: (4, 1.5) = (1.5, 4)
2. Reflection over the y-axis: (x, y) → (-x, y)
Final rotated endpoint 2: (-1.5, 4)
Now, we can calculate the length of the new segment using the new coordinates of the endpoints.
Length of the new segment = √[(-1.5 - (-4))^2 + (4 - 0)^2]
= √[(-1.5 + 4)^2 + 4^2]
= √[2.5^2 + 4^2]
= √[6.25 + 16]
= √22.25
≈ 4.71
Therefore, the length of the segment after rotating 270° clockwise is approximately 4.71 units.
wrong dumb bot
Apologies for the previous response. Let's correct the mistake and find the correct length of the segment after rotating 270° clockwise.
To rotate a point (x, y) 270° clockwise, we can use the following formula:
New_x = x * cos(270°) - y * sin(270°)
New_y = x * sin(270°) + y * cos(270°)
Let's apply this formula to the endpoints of the segment.
Endpoint 1: (0, 4)
New_x1 = 0 * cos(270°) - 4 * sin(270°) = 0 - (-4) = 4
New_y1 = 0 * sin(270°) + 4 * cos(270°) = 0 + 0 = 0
Endpoint 2: (3, 4)
New_x2 = 3 * cos(270°) - 4 * sin(270°) = 0 - (-4) = 4
New_y2 = 3 * sin(270°) + 4 * cos(270°) = -3 + 0 = -3
The rotated endpoints are (4, 0) and (4, -3).
Now, let's calculate the length of the new segment using the coordinates of the rotated endpoints.
Length of the new segment = √((4 - 4)^2 + (0 - (-3))^2)
= √(0^2 + 3^2)
= √(0 + 9)
= √9
= 3
Therefore, the length of the segment after rotating 270° clockwise is 3 units.