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select the correct answer. quadrilateral abcd underwent a sequence of t…

Question

select the correct answer.

quadrilateral abcd underwent a sequence of transformations to give quadrilateral abcd. which transformations could have taken place?

a reflection across the x-axis followed by a reflection across the y-axis

a translation 10 units down followed by a translation 8 units to the right

a rotation 90° counterclockwise about the origin followed by a reflection across the y-axis

a reflection across the line y = x followed by a reflection across the line y = -x

Explanation:

Identify coordinates of the original and transformed vertices

The vertices of quadrilateral \(ABCD\) are:

$$ A(-6, 4), \quad B(-6, 6), \quad C(-2, 6), \quad D(-4, 4) $$

The vertices of the transformed quadrilateral \(A'B'C'D'\) are:

$$ A'(1, -6), \quad B'(1, -4), \quad C'(5, -4), \quad D'(3, -6) $$

Test the given transformation options

Let's evaluate the transformation options on a representative point, say \(B(-6, 6)\), which must map to \(B'(1, -4)\):

  1. Option 1: Reflection across the x-axis followed by a reflection across the y-axis
  • Reflection across x-axis: \((x, y) \to (x, -y)\). For \(B(-6, 6) \to (-6, -6)\).
  • Reflection across y-axis: \((x, -y) \to (-x, -y)\). For \((-6, -6) \to (6, -6)\).
  • This does not match \(B'(1, -4)\).
  1. Option 2: Translation 10 units down followed by a translation 8 units to the right
  • Translation 10 units down: \((x, y) \to (x, y - 10)\). For \(B(-6, 6) \to (-6, -4)\).
  • Translation 8 units to the right: \((x, y - 10) \to (x + 8, y - 10)\). For \((-6, -4) \to (2, -4)\).
  • This does not match \(B'(1, -4)\).
  1. Option 3: Rotation \(90^\circ\) counterclockwise about the origin followed by a reflection across the y-axis
  • Rotation \(90^\circ\) counterclockwise: \((x, y) \to (-y, x)\). For \(B(-6, 6) \to (-6, -6)\).
  • Reflection across y-axis: \((-y, x) \to (y, x)\). For \((-6, -6) \to (6, -6)\).
  • This does not match \(B'(1, -4)\).
  1. Option 4: Reflection across the line \(y = -x\) followed by a reflection across the line \(y = x\)
  • Let's check the standard coordinate rules:
  • Reflection across \(y = -x\): \((x, y) \to (-y, -x)\).

For \(B(-6, 6) \to (-6, 6)\).

  • Reflection across \(y = x\): \((-y, -x) \to (-x, -y)\).

For \((-6, 6) \to (-6, 6)\).

  • This does not match \(B'(1, -4)\).

Let's re-examine the coordinates of \(A'B'C'D'\) from the graph:

  • \(A'\) is at \((1, -6)\)
  • \(B'\) is at \((1, -4)\)
  • \(C'\) is at \((5, -4)\)
  • \(D'\) is at \((3, -6)\)

Let's re-evaluate Option 3:

  • Rotation \(90^\circ\) counterclockwise: \((x, y) \to (-y, x)\).

For \(B(-6, 6) \to (-6, -6)\).
Wait, let's test \(A(-6, 4)\):

  • Rotation \(90^\circ\) counterclockwise: \((-4, -6)\).
  • Reflection across the y-axis: \((4, -6)\). This does not match \(A'(1, -6)\).

Let's re-read Option 3: "a rotation 90° counterclockwise about the origin followed by a reflection across the y-axis".
Wait, let's look at the options again. Is there a translation option?
"a translation 10 units down followed by a translation 8 units to the right"
Wait, let's look at the coordinates of \(B'\) again.
The y-axis has grid lines. Let's count carefully:
The origin is \((0,0)\).
To the right: \(2, 4, 6, 8\).
The point \(B'\) is at \(x = 1\)? No, let's look at the grid.
The grid line for \(2\) is labeled. The line to the left of \(2\) is \(1\)? No, the grid lines are spaced by 1 unit.
Let's count from the y-axis (0) to the right:
1st grid line = 1
2nd grid line = 2 (labeled)
3rd grid line = 3
4th grid line = 4 (labeled)
So \(B'\) is on the grid line \(x = 2\).
Let's check the coordinates of \(B'\) with \(x = 2\):
If \(B'\)…

Answer:

  • a reflection across the x-axis followed by a reflection across the y-axis
  • a translation 10 units down followed by a translation 8 units to the right (Correct answer)
  • a rotation 90° counterclockwise about the origin followed by a reflection across the y-axis
  • a reflection across the line y = -x followed by a reflection across the line y = x