Rectangle STUV is shown on a coordinate plane.
Rectangle STUV with vertices S at negative 7 comma 6, T at negative 2 comma 6, U at negative 2 comma 1, and V at negative 7 comma 1.
If rectangle STUV is translated using the rule (x, y) → (x − 2, y − 4) and then rotated 90° counterclockwise, what is the location of S″?
(3, −9)
(3, −4)
(−2, −4)
(−2, −9)
The location of S″ after the translation and rotation can be found by first applying the translation rule (x, y) → (x − 2, y − 4) to the coordinates of point S (-7, 6):
S' = (-7 - 2, 6 - 4) = (-9, 2)
Next, to rotate the point S' 90° counterclockwise, we can use the rotation matrix formula:
[x', y'] = [0 -1; 1 0] * [x; y]
Plugging in the values of S' (-9, 2):
S'' = [0 -1; 1 0] * [-9; 2] = [(-9 * 0) + (-1 * 2); (1 * -9) + (0 * 2)] = (-2, -9)
Therefore, the location of S'' after the translation and rotation is (-2, -9). Therefore, the correct answer is (−2, −9).