Question

which segment is perpendicular to \\(\overline{eb}\\)?

use for #1-4

\\(\bigcirc\\) \\(\overline{fc}\\)
\\(\bigcirc\\) \\(\overline{bc}\\)
\\(\bigcirc\\) \\(\overline{df}\\)
\\(\bigcirc\\) \\(\overline{ac}\\)

Explanation:

⚡ Using what you learned: Three-Dimensional Shapes & Solids

Step 1: Identify the 3D shape and its components

The given figure is a right triangular prism.

• The two triangular bases are \(\triangle ABC\) at the top and \(\triangle DEF\) at the bottom.
• The vertical edges connecting the two bases are \(\overline{AD}\), \(\overline{BE}\), and \(\overline{CF}\). These vertical edges are perpendicular to both the top and bottom triangular bases.

Step 2: Analyze the relationship with segment EB

The segment \(\overline{EB}\) (or \(\overline{BE}\)) is a vertical lateral edge of the prism.

In a right prism, any lateral edge is perpendicular to every line in the base plane that passes through its intersection point:

• At the top base, \(\overline{EB}\) is perpendicular to the plane containing \(\triangle ABC\). Therefore, \(\overline{EB}\) is perpendicular to \(\overline{AB}\) and \(\overline{BC}\).
• At the bottom base, \(\overline{EB}\) is perpendicular to the plane containing \(\triangle DEF\). Therefore, \(\overline{EB}\) is perpendicular to \(\overline{DE}\) and \(\overline{EF}\).

Step 3: Match with the given options

Let's evaluate the given choices to see which segment is perpendicular to \(\overline{EB}\):

• \(\overline{FC}\): This is another vertical lateral edge, which is parallel to \(\overline{EB}\), not perpendicular.
• \(\overline{BC}\): This segment lies in the top base plane and meets the vertical edge \(\overline{EB}\) at point \(B\). Since the lateral edge is perpendicular to the base plane, \(\overline{BC}\) is perpendicular to \(\overline{EB}\).
• \(\overline{DF}\): This is a segment in the bottom base, but it does not meet \(\overline{EB}\) directly to form a right angle in this context.
• \(\overline{AC}\): This is a segment in the top base, but it does not intersect \(\overline{EB}\).

Thus, \(\overline{BC}\) is perpendicular to \(\overline{EB}\).

Answer:

\(\overline{BC}\)