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Zorluk: Çok zorTransformations in the Coordinate Plane

On a coordinate plane, triangle PQRPQR has vertices P(2,3)P(2, 3), Q(5,3)Q(5, 3), and R(2,7)R(2, 7). The triangle undergoes a sequence of three transformations:

1. A dilation centered at the point C(1,1)C(1, 1) with a scale factor of 22.
2. A reflection across the line y=−xy = -x.
3. A translation of 33 units to the left and 44 units up.

What are the coordinates of the final image of vertex QQ after this sequence of transformations?

  1. A
    (−9,−6)(-9, -6)
  2. B
    (2,13)(2, 13)
  3. C
    (−2,−13)(-2, -13)
  4. (−8,−5)(-8, -5)Cevap
  5. E
    (−7,−4)(-7, -4)

Cevap

The final image of vertex QQ is located at the coordinates (−8,−5)(-8, -5).
Applying the transformations sequentially yields the correct result. First, the vector from the dilation center at (1,1)(1, 1) to (5,3)(5, 3) is (4,2)(4, 2). Scaling this vector by 22 gives (8,4)(8, 4), which when added back to (1,1)(1, 1) places the intermediate image at (9,5)(9, 5). Second, reflecting (9,5)(9, 5) across the line y=−xy = -x yields (−5,−9)(-5, -9). Third, translating this point 33 units left and 44 units up results in (−5−3,−9+4)=(−8,−5)(-5 - 3, -9 + 4) = (-8, -5).

Adım Adım Çözüm

1
Find the vector from the center of dilation C(1,1)C(1, 1) to the point Q(5,3)Q(5, 3).
Vector CQ⃗=(5−1,3−1)=(4,2)\vec{CQ} = (5 - 1, 3 - 1) = (4, 2).
To perform a dilation centered at a point other than the origin, we must first find the displacement of the target point relative to that center.
2
Multiply the vector by the scale factor of 22 and add it back to the coordinates of the center C(1,1)C(1, 1).
First intermediate point Q′=(1,1)+2(4,2)=(1+8,1+4)=(9,5)Q' = (1, 1) + 2(4, 2) = (1 + 8, 1 + 4) = (9, 5).
This scales the distance from the center of dilation by 22 and finds the coordinate of the image point Q′Q'.
3
Apply the reflection rule for the line y=−xy = -x to the point Q′(9,5)Q'(9, 5).
Second intermediate point Q′′=(−5,−9)Q'' = (-5, -9).
Reflecting a coordinate (x,y)(x, y) across the line y=−xy = -x swaps and negates both coordinates, mapping (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x).
4
Translate the point Q′′(−5,−9)Q''(-5, -9) by subtracting 33 from the x-coordinate and adding 44 to the y-coordinate.
Final point Q′′′=(−5−3,−9+4)=(−8,−5)Q''' = (-5 - 3, -9 + 4) = (-8, -5).
Translating left reduces the x-value, and translating up increases the y-value.

Anahtar Kavram

Composite transformations in the coordinate plane including non-origin dilations, reflections, and translations.
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Konular

Transformations in the Coordinate Plane

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