Examkin

Question

Difficulty: HardTransformations in the Coordinate Plane

A square PQRSPQRS has vertices at P(2,2)P(2, 2), Q(6,2)Q(6, 2), R(6,6)R(6, 6), and S(2,6)S(2, 6) in the coordinate plane. The square is first rotated 90∘90^\circ counterclockwise about the origin, and then translated such that the final image of vertex RR is located at (1,5)(1, 5). What are the coordinates of the final image of vertex PP after this sequence of transformations?

  1. A
    (−3,9)(-3, 9)
  2. (5,1)(5, 1)Answer
  3. C
    (−9,3)(-9, 3)
  4. D
    (5,9)(5, 9)
  5. E
    (−3,1)(-3, 1)

Answer

(5,1)(5, 1)
To find the coordinates of the final image of vertex P, we perform the transformations step-by-step. First, a 90∘90^\circ counterclockwise rotation about the origin maps any point (x,y)(x, y) to (−y,x)(-y, x). Under this rotation, P(2,2)P(2, 2) maps to P′(−2,2)P'(-2, 2) and R(6,6)R(6, 6) maps to R′(−6,6)R'(-6, 6). Second, we find the translation vector by comparing the rotated vertex R′(−6,6)R'(-6, 6) to its final position R′′(1,5)R''(1, 5). The change in the x-coordinate is 1−(−6)=71 - (-6) = 7, and the change in the y-coordinate is 5−6=−15 - 6 = -1. This represents a translation of 77 units right and 11 unit down, or the vector (7,−1)(7, -1). Finally, applying this translation to the intermediate point P′(−2,2)P'(-2, 2) gives P′′(−2+7,2−1)=(5,1)P''(-2 + 7, 2 - 1) = (5, 1).

Step-by-Step Solution

1
Apply the 90∘90^\circ counterclockwise rotation about the origin to the vertices P(2,2)P(2, 2) and R(6,6)R(6, 6).
Using the rotation rule (x,y)→(−y,x)(x, y) \rightarrow (-y, x), vertex P(2,2)P(2, 2) maps to P′(−2,2)P'(-2, 2) and vertex R(6,6)R(6, 6) maps to R′(−6,6)R'(-6, 6).
This establishes the intermediate positions of the vertices before the translation is applied.
2
Determine the translation vector (h,k)(h, k) that maps the intermediate point R′(−6,6)R'(-6, 6) to the final point R′′(1,5)R''(1, 5).
Solving the equations −6+h=1-6 + h = 1 and 6+k=56 + k = 5 gives h=7h = 7 and k=−1k = -1. The translation vector is (7,−1)(7, -1).
The translation shifts all points in the plane by the same distance and direction.
3
Apply the translation vector (7,−1)(7, -1) to the intermediate vertex P′(−2,2)P'(-2, 2) to find the final image P′′P''.
P′′=(−2+7,2−1)=(5,1)P'' = (-2 + 7, 2 - 1) = (5, 1).
Applying the same translation to the rotated position of vertex P yields its final coordinates.

Key Concept

Composite transformations in the coordinate plane
Estimated Time:2m 0s
Rate this question

Topics

Transformations in the Coordinate Plane

More questions from this topic