Question
If a segment has endpoints (0, 4) and (3, 4) , how long will the segment be after it is rotated 270° clockwise?(1 point)
To determine the length of the segment after rotating it 270° clockwise, we need to find the distance between the rotated endpoints.
First, let's plot the original segment on a coordinate plane:
Segment AB with endpoints A(0, 4) and B(3, 4)
Next, let's rotate the endpoints 270° clockwise. To do this, we can use the rotation formula for a point (x, y) with angle θ:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
For the first endpoint (0, 4):
x' = 0*cos(270°) - 4*sin(270°) = 0 + 4 = 4
y' = 0*sin(270°) + 4*cos(270°) = 0 - 4 = -4
So the rotated endpoint is (4, -4)
For the second endpoint (3, 4):
x' = 3*cos(270°) - 4*sin(270°) = 0 - 4 = -4
y' = 3*sin(270°) + 4*cos(270°) = -3 + 0 = -3
So the rotated endpoint is (-4, -3)
Now, we can find the length of the rotated segment using the distance formula:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
Length = √[(-4 - 4)^2 + (-3 - (-4))^2]
Length = √[(-8)^2 + (1)^2]
Length = √[64 + 1]
Length = √65
Therefore, the segment will be √65 units long after it is rotated 270° clockwise.