If a segment has endpoint (0,4) and (3,4) how long will the segment be after its rotated 290° clockwise?
To find the length of the segment after it is rotated 290° clockwise, we need to find the new coordinates of the endpoints.
First, let's find the coordinates of the midpoint of the segment. Since the y-coordinate is the same for both endpoints, the midpoint will also have a y-coordinate of 4. To find the x-coordinate, we can take the average of the x-coordinates of the endpoints: (0 + 3) / 2 = 1.5.
Now, let's rotate the midpoint 290° clockwise. To do this, we can use the rotation formula:
x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)
where (x, y) are the original coordinates, (x', y') are the new coordinates, and theta is the angle of rotation.
In this case, our midpoint is at (1.5, 4), and we want to rotate it 290° clockwise. Plugging in the values into the rotation formula:
x' = 1.5 * cos(290°) - 4 * sin(290°)
y' = 1.5 * sin(290°) + 4 * cos(290°)
Using a calculator, we can find the approximate values of:
x' ≈ -0.26
y' ≈ 3.88
Now, we can find the length of the segment from (0, 4) to (-0.26, 3.88) using the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-0.26 - 0)^2 + (3.88 - 4)^2)
≈ sqrt(0.0676 + 0.0016)
≈ sqrt(0.0692)
≈ 0.263
Therefore, the length of the segment after it is rotated 290° clockwise is approximately 0.263 units.