Line segment RS is shown on a coordinate grid:
A coordinate grid is shown from positive 6 to negative 6 on the x-axis and from positive 6 to negative 6 on the y-axis. A line segment RS is shown with S as ordered pair negative 1, negative 3 and R as ordered pair negative 5, negative 3.
The line segment is rotated 270 degrees counterclockwise about the origin to form R′S′. Which statement describes R′S′? (5 points)
R′S′ is parallel to RS.
R′S′ is half the length of RS.
R′S′ is twice the length of RS.
R′S′ is equal in length to RS.
To rotate a point (x, y) counterclockwise about the origin (0, 0) by 270 degrees, we can use the following transformation:
(x', y') = (-y, x)
For the point R(-5, -3), applying the transformation:
R' = (-(-3), -5)
= (3, -5)
For the point S(-1, -3), applying the transformation:
S' = (-(-3), -1)
= (3, -1)
So, the new line segment R'S' is formed by connecting the points (3, -5) and (3, -1).
To determine the length of R'S', we can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((3 - 3)^2 + (-1 - (-5))^2)
= sqrt(0^2 + 4^2)
= sqrt(16)
= 4
The length of R'S' is 4.
Therefore, the correct statement is:
R′S′ is equal in length to RS.