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)
A. R′S′ is parallel to RS.
B. R′S′ is half the length of RS.
C. R′S′ is twice the length of RS.
D. R′S′ is equal in length to RS.
To rotate the line segment 270 degrees counterclockwise about the origin, we can apply the following transformation to each point:
(x, y) -> (-y, x)
Applying this transformation to R(-5, -3), we get R'(-3, -5). Applying the same transformation to S(-1, -3), we get S'(-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 = √((x2 - x1)^2 + (y2 - y1)^2)
Using R'(-3, -5) as (x1, y1) and S'(-3, 1) as (x2, y2):
d = √((-3 - (-3))^2 + (1 - (-5))^2)
= √(0^2 + 6^2)
= √(0 + 36)
= √36
= 6
Since the length of RS is also 6, the correct statement is:
D. R'S' is equal in length to RS.