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Difficulty: MediumTransformations in the Coordinate Plane

A line segment with endpoints C(2,3)C(2, 3) and D(6,3)D(6, 3) is plotted in the standard (x,y)(x, y) coordinate plane. If this segment is rotated 90∘90^\circ clockwise about the origin and then translated 44 units up, what are the coordinates of the midpoint of the resulting segment?

  1. A
    (3,−8)(3, -8)
  2. B
    (7,−4)(7, -4)
  3. (3,0)(3, 0)Answer
  4. D
    (−3,8)(-3, 8)
  5. E
    (4,7)(4, 7)

Answer

(3,0)(3, 0)
The correct answer is obtained by first calculating the midpoint of the original segment CDCD, which yields (4,3)(4, 3). Applying a 90∘90^\circ clockwise rotation about the origin maps the point (x,y)(x, y) to (y,−x)(y, -x), transforming (4,3)(4, 3) to (3,−4)(3, -4). Translating this intermediate point 4 units up adds 4 to its y-coordinate, resulting in (3,0)(3, 0).

Step-by-Step Solution

1
Find the midpoint of the original line segment CDCD using the midpoint formula M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).
M=(2+62,3+32)=(4,3)M = \left(\frac{2 + 6}{2}, \frac{3 + 3}{2}\right) = (4, 3)
Since translations and rotations are rigid transformations, the midpoint of the transformed segment is the image of the midpoint of the original segment.
2
Apply the rotation of 90∘90^\circ clockwise about the origin to the midpoint M(4,3)M(4, 3). The rule for a clockwise rotation of 90∘90^\circ is (x,y)→(y,−x)(x, y) \rightarrow (y, -x).
M′(3,−4)M'(3, -4)
Rotating a point (x,y)(x, y) by 90∘90^\circ clockwise maps it to (y,−x)(y, -x).
3
Translate the point M′(3,−4)M'(3, -4) by 44 units up. The rule for translating a point dd units up is (x,y)→(x,y+d)(x, y) \rightarrow (x, y + d).
M′′(3,−4+4)=(3,0)M''(3, -4 + 4) = (3, 0)
A vertical translation upward increases the y-coordinate of the point by the given number of units.

Key Concept

Transformations in the Coordinate Plane
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Topics

Transformations in the Coordinate Plane

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