If a segment has endpoint (0,4) and (3,4), how long will the segment be after its rotated 290° clockwise? How many units?
To find the length of the segment after rotating it 290° clockwise, we need to find the distance between the new endpoints after rotation.
First, let's find the new coordinates of the endpoints after rotating 290° clockwise.
The formula for rotating a point (x, y) about the origin (0, 0) by an angle θ is:
x’ = x*cos(θ) - y*sin(θ)
y’ = x*sin(θ) + y*cos(θ)
For the first endpoint (0, 4):
x' = 0*cos(290°) - 4*sin(290°)
y' = 0*sin(290°) + 4*cos(290°)
Using a calculator or trigonometric table to find the sine and cosine values for 290°, we get:
x' ≈ 0*cos(290°) - 4*(-0.156434)
≈ -0.625736
y' ≈ 0*sin(290°) + 4*0.987688
≈ 3.950752
Similarly, for the second endpoint (3, 4):
x' ≈ 3*cos(290°) - 4*sin(290°)
y' ≈ 3*sin(290°) + 4*cos(290°)
Again, using the values from the previous step:
x' ≈ 3*(-0.156434) - 4*(-0.156434)
≈ -2.657303
y' ≈ 3*0.987688 + 4*0.987688
≈ 7.439372
Now, let's find the distance between the new endpoints using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((-2.657303 - (-0.625736))^2 + (7.439372 - 3.950752)^2)
Calculating the above expression, we get:
Distance ≈ √((-2.657303 + 0.625736)^2 + (7.439372 - 3.950752)^2)
≈ √((-2.031567)^2 + (3.488620)^2)
≈ √(4.127660 + 12.149923)
≈ √(16.277583)
≈ 4.03
Therefore, the length of the segment after rotating 290° clockwise is approximately 4.03 units.