if a segment has endpoints (0, 4) and (3, 4), how long will the segment be after it is rotated 270 degrees clockwise?
To find the length of the segment after it is rotated 270 degrees clockwise, we need to calculate the distance between the new endpoints.
Let's calculate the new coordinates of the endpoints after rotating 270 degrees clockwise:
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
For the second endpoint (3, 4):
x'' = 3*cos(270) - 4*sin(270) = 0 + 0 = 0
y'' = 3*sin(270) + 4*cos(270) = -3 + 0 = -3
So, the new endpoints after the rotation are (4, -4) and (0, -3).
To calculate the length of the segment, we can use the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
distance = √((0 - 4)^2 + (-3 - (-4))^2)
= √((-4)^2 + (1)^2)
= √(16 + 1)
= √17
Therefore, the length of the segment after rotating 270 degrees clockwise is √17.
thats wrong
Apologies for the incorrect response. Let's calculate the correct length of the segment after it is rotated 270 degrees clockwise.
The original segment has endpoints (0, 4) and (3, 4).
To rotate a point (x, y) 270 degrees clockwise about the origin, we can use the following rotation formulas:
x' = x*cos(270) - y*sin(270)
y' = x*sin(270) + y*cos(270)
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
For the second endpoint (3, 4):
x'' = 3*cos(270) - 4*sin(270) = 0 + 0 = 0
y'' = 3*sin(270) + 4*cos(270) = -3 + 0 = -3
So, the new endpoints after rotating 270 degrees clockwise are (4, -4) and (0, -3).
Now, we can calculate the length of the segment by using the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
distance = √((0 - 4)^2 + (-3 - (-4))^2)
= √((-4)^2 + (1)^2)
= √(16 + 1)
= √17
Therefore, the correct length of the segment after rotating 270 degrees clockwise is √17.
if a segment has endpoints (0, 4) and (3, 4), how long will the segment be after it is rotated 270 degrees clockwise?
__units
Apologies for the confusion caused by my previous incorrect responses. I appreciate your patience.
To determine the length of the segment after it is rotated 270 degrees clockwise, let's consider the original endpoints (0, 4) and (3, 4).
When a point is rotated 270 degrees clockwise about the origin, the new coordinates can be found using the following rotation formulas:
x' = x*cos(270) - y*sin(270)
y' = x*sin(270) + y*cos(270)
Applying these formulas to 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
For the second endpoint (3, 4):
x'' = 3*cos(270) - 4*sin(270) = 0 + 0 = 0
y'' = 3*sin(270) + 4*cos(270) = -3 + 0 = -3
Therefore, the new endpoints after rotating 270 degrees clockwise are (4, -4) and (0, -3).
To determine the length of the segment, we can once again use the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
distance = √((0 - 4)^2 + (-3 - (-4))^2)
= √((-4)^2 + (1)^2)
= √(16 + 1)
= √17
Thus, the length of the segment after rotating 270 degrees clockwise is √17 units.
17 is not the ANSWER
Apologies, I made an error in my previous response. Let's correct it.
Given the endpoints of the segment as (0, 4) and (3, 4), we need to find the length of the segment after rotating it 270 degrees clockwise.
When a point is rotated 270 degrees clockwise about the origin, the new coordinates (x', y') can be found using the following rotation formulas:
x' = x*cos(270) - y*sin(270)
y' = x*sin(270) + y*cos(270)
For the first endpoint (0, 4):
x' = 0*cos(270) - 4*sin(270) = 0 - 4 = -4
y' = 0*sin(270) + 4*cos(270) = 0 + 0 = 0
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
The new endpoints after rotating 270 degrees clockwise are (-4, 0) and (-4, -3).
To calculate the length of the segment, we once again use the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
distance = √((-4 - (-4))^2 + (-3 - 0)^2)
= √(0^2 + (-3)^2)
= √(0 + 9)
= √9
= 3
Therefore, the length of the segment after rotating 270 degrees clockwise is 3 units.