if a segment has endpoints (0,4) and (3,4) how long will the segment be after it rotated 120 degrees clockwise?
To find the length of the segment after rotating 120 degrees clockwise, we need to determine the new coordinates of the endpoints first.
The segment has endpoints (0,4) and (3,4). The midpoint of the segment can be found by taking the average of the x-coordinates and the y-coordinates:
Midpoint = ( (0+3)/2 , (4+4)/2 ) = (1.5, 4)
To rotate a point 120 degrees clockwise, we can use the rotation formula:
x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)
where theta is the angle of rotation.
Let's rotate the midpoint of the segment, (1.5, 4), by 120 degrees clockwise:
x' = 1.5 * cos(120) - 4 * sin(120)
y' = 1.5 * sin(120) + 4 * cos(120)
Using the values from a unit circle or trigonometric ratios, we can simplify the equations:
x' = 1.5 * (-0.5) - 4 * (√3 / 2) = -0.75 - 2√3
y' = 1.5 * (√3 / 2) + 4 * (-0.5) = 0.75 - 2
The new endpoint is (-0.75 - 2√3, 0.75 - 2).
To find the length of the segment, we use the distance formula between the two points:
Length = √ [ (x2 - x1)^2 + (y2 - y1)^2 ]
= √ [ (-0.75 - 2√3 - 0)^2 + (0.75 - 2 - 4)^2 ]
You can simplify the expression further by plugging in the values and performing the calculations.