Trapezoid ABCD coordinates: A(3,1), B(7,1) C(6,4)D(4,4)

Question

Trapezoid A'B'C'D' coordinates: A'(-1,3) B'(-1,7) C'(-4,6) D'(-4,4)
Which explanation for the congruency of the trapezoids is correct?
Statement #1: The trapezoids are congruent because ABCD was reflected over the line y=z
Statement #2: The trapezoids are congruent because ABCD was translated left 4 units and up 2 units
Statement #3: The trapezoids are congruent because ABCD was rotated 270 degrees (counterclockwise)
Statement #4: The trapezoids are congruent because ABCD was rotated 90 degrees (counterclockwise)
Which statement is correct?

Answer 1

To determine which statement is correct about the congruency of trapezoids ABCD and A'B'C'D', let's analyze the coordinates:

• Trapezoid ABCD:

  • A(3, 1)
  • B(7, 1)
  • C(6, 4)
  • D(4, 4)

• Trapezoid A'B'C'D':

  • A'(-1, 3)
  • B'(-1, 7)
  • C'(-4, 6)
  • D'(-4, 4)

To compare the coordinates, let's test each statement to see if they correctly describe a transformation that leads from ABCD to A'B'C'D'.

Statement #1: Reflection over the line y = z.
When reflecting a point (x, y) over the line y = x, the new coordinates become (y, x). Applying this transformation:

• A(3, 1) becomes (1, 3)
• B(7, 1) becomes (1, 7)
• C(6, 4) becomes (4, 6)
• D(4, 4) becomes (4, 4)
  This does not match A', B', C', and D'.

Statement #2: Translation left 4 units and up 2 units.
Translating points by subtracting 4 from the x-coordinate and adding 2 to the y-coordinate:

• A(3, 1) => A'(-1, 3)
• B(7, 1) => B'(-1, 7)
• C(6, 4) => C'(-4, 6)
• D(4, 4) => D'(-4, 4)
  This matches all the points exactly. This statement seems to be correct.

Statement #3: Rotation 270 degrees (counterclockwise).
Rotating 270 degrees (or 90 degrees clockwise) would not yield the same points as A'B'C'D'.

Statement #4: Rotation 90 degrees (counterclockwise).
When rotating a point (x, y) 90 degrees counterclockwise, the new coordinates become (-y, x). Applying to ABCD:

• A(3, 1) becomes (-1, 3)
• B(7, 1) becomes (-1, 7)
• C(6, 4) becomes (-4, 6)
• D(4, 4) becomes (-4, 4)
  This matches A', B', C', D'.

Therefore, both Statement #2 (translation left 4 units and up 2 units) and Statement #4 (rotation 90 degrees counterclockwise) correctly describe transformations that result in congruent trapezoids.

However, Statement #2 clearly is a simpler and more straightforward way to reach from ABCD to A'B'C'D', as it involves a direct translation rather than a rotation.

So the correct conclusion is that Statement #2 is one correct explanation for the congruency of the trapezoids.