Question

of the four options, which line segment is perpendicular to \\( \overline { h e } \\)?
a \\( \overline { d e } \\)
b \\( \overline { g f } \\)
c \\( \overline { a g } \\)
d \\( \overline { b d } \\)

Explanation:

Step1: Recall the definition of perpendicular lines

Perpendicular lines are lines that intersect at a right angle ($90^{\circ}$).

Step2: Analyze each option

• Option A: $\overline{DE}$

$\overline{DE}$ is parallel to $\overline{HE}$ (in a rectangular - prism, opposite edges of a face are parallel). So, $\overline{DE}$ is not perpendicular to $\overline{HE}$.

• Option B: $\overline{GF}$

$\overline{GF}$ is parallel to $\overline{AC}$ and also parallel to $\overline{BD}$. Since $\overline{HE}$ is in a different orientation (in a three - dimensional figure), $\overline{GF}$ is not perpendicular to $\overline{HE}$.

• Option C: $\overline{AG}$

$\overline{AG}$ is parallel to $\overline{DH}$ (in a rectangular - prism, opposite edges are parallel). $\overline{AG}$ is not perpendicular to $\overline{HE}$.

• Option D: $\overline{BD}$

In a rectangular - prism (assuming the base is a rectangle), if we consider the base $BDEH$ (a rectangle), in a rectangle, adjacent sides are perpendicular. So, $\overline{BD}\perp\overline{HE}$

Answer:

D. $\overline{BD}$