Ask a New Question
Menu

Use the graph to answer the question.

Graph of polygon ABCDE with vertices at negative 1 comma negative 4, negative 1 comma negative 1, 3 comma negative 1, 3 comma negative 4, 1 comma negative 6. A second polygon A prime B prime C prime D prime E prime with vertices at negative 11 comma negative 4, negative 11 comma negative 1, negative 15 comma negative 1, negative 15 comma negative 4, negative 13 comma negative 6.

Determine the line of reflection.
Reflection across the x-axis
Reflection across x = −6
Reflection across the y-axis
Reflection across y = −6

1 answer

To determine the line of reflection that maps polygon ABCDE to polygon A'B'C'D'E', we can analyze the given vertices for both polygons.

Vertices of polygon ABCDE:

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

Vertices of polygon A'B'C'D'E':

  • A'(-11, -4)
  • B'(-11, -1)
  • C'(-15, -1)
  • D'(-15, -4)
  • E'(-13, -6)

Next, we find the transformation required to move points from ABCDE to A'B'C'D'E'.

Observing the x-coordinates:

  • A(-1) to A'(-11): The x-coordinate changes from -1 to -11, which is a shift of -10.
  • B(-1) to B'(-11): Same as A, shift of -10.
  • C(3) to C'(-15): This is a shift from 3 to -15, equal to -18.
  • D(3) to D'(-15): Same as C, shift of -18.
  • E(1) to E'(-13): This is a shift from 1 to -13, equal to -14.

The shifts are inconsistent across points, indicating that a simple translation or rotation does not apply. Next, we look for a reflection line in the x-y plane.

Now, we can assess potential reflection lines:

  1. Reflection across the x-axis: This would negate the y-coordinates, but it doesn’t relate to the x-coordinates of A to A' or the other corresponding vertices.

  2. Reflection across the y-axis: This would negate the x-coordinates. However, this does not account for the shifts observed from ABCDE to A'B'C'D'E'.

  3. Reflection across a vertical line x = -6: We calculate how points would reflect over this line.

The formula for reflection over the line x = a is given by:

  • for a point (x, y), the reflection is (2a - x, y).

Using the line x = -6:

  1. A(-1, -4) reflects to:

    • x = 2(-6) - (-1) = -12 + 1 = -11 Thus, A'(-11, -4)

  2. B(-1, -1) reflects to:

    • x = 2(-6) - (-1) = -12 + 1 = -11 Thus, B'(-11, -1)

  3. C(3, -1) reflects to:

    • x = 2(-6) - (3) = -12 - 3 = -15 Thus, C'(-15, -1)

  4. D(3, -4) reflects to:

    • x = 2(-6) - (3) = -12 - 3 = -15 Thus, D'(-15, -4)

  5. E(1, -6) reflects to:

    • x = 2(-6) - (1) = -12 - 1 = -13 Thus, E'(-13, -6)

Since all corresponding points reflect correctly, the line of reflection that maps ABCDE to A'B'C'D'E' is:

Reflection across x = -6.