Question

a figure is graphed in a coordinate plane with vertices at d(8, -3), u(-3, 5), k(2, 7), and e(7, -6). the figure is then reflected across the x - axis to form the image duke. which statement is true?

• $overline{ed}$ is parallel to $overline{ku}$.
• $overline{uk}$ is parallel to $overline{ek}$.
• $overline{ke}$ is parallel to $overline{de}$.
• $overline{du}$ is parallel to $overline{de}$

Explanation:

Step1: Recall Reflection Properties

Reflection is a rigid transformation, so the image (after reflection) is congruent to the pre - image, and corresponding sides are parallel (if the reflection is over a horizontal or vertical line, we can analyze the slopes of the sides). First, we need to assume the original vertices (even though some are partially visible, we can infer the shape). Let's assume the original figure has vertices that form a quadrilateral (since there are four vertices: D, U, K, E and their images D', U', K', E'). After reflection over the x - axis (assuming the reflection is over x - axis as it's a common case), the y - coordinates of the points are negated, but the x - coordinates remain the same (or vice - versa for y - axis, but let's proceed with x - axis for example).

Step2: Analyze Side Directions

For a quadrilateral, in the pre - image, if we consider the sides:

• Let's analyze the first option: $\overline{E'D'}$ and $\overline{K'U'}$. If we assume that in the original figure, $\overline{ED}$ and $\overline{KU}$ are parallel (since reflection preserves the direction of horizontal or vertical sides), then after reflection, their images $\overline{E'D'}$ and $\overline{K'U'}$ will also be parallel.
• For the second option: $\overline{U'K'}$ and $\overline{E'K'}$: $\overline{U'K'}$ and $\overline{E'K'}$ meet at $K'$, so they are not parallel (they are adjacent sides of the quadrilateral, so they intersect at a vertex).
• For the third option: $\overline{K'E'}$ and $\overline{D'E'}$: These two sides meet at $E'$, so they are adjacent sides and not parallel.
• For the fourth option: $\overline{D'U'}$ and $\overline{D'E'}$: These two sides meet at $D'$, so they are adjacent sides and not parallel.

Answer:

$\boldsymbol{\overline{E'D'}\text{ is parallel to }\overline{K'U'}}$ (the first option among the four purple - boxed options)