Identifying the Diagonal in a Rectangular Prism: Line Segment Analysis

Space Diagonals with Prism Vertices

Identify which of the following line segments is the diagonal of the rectangular prism?

AAABBBCCCDDDEEEFFFGGGHHH

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Step-by-step video solution

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00:00 Which of these segments is the cube's diagonal?
00:03 The cube's diagonal connects 2 vertices that are not on the same face
00:07 There are 4 such diagonals in each cube and they are all equal to each other
00:15 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

Identify which of the following line segments is the diagonal of the rectangular prism?

AAABBBCCCDDDEEEFFFGGGHHH

2

Step-by-step solution

A box diagonal is a diagonal that passes between two vertices that are not connected,
meaning, the diagonal passes through the box, and not on one of its edges.

Let's see which of the segments in the answers passes entirely through the box from end to end.

AC, HF and FC are all diagonals that lie on the edges of the box, therefore the segment that fits this description is BH.

3

Final Answer

BH BH

Key Points to Remember

Essential concepts to master this topic
  • Definition: Space diagonal connects opposite vertices through the interior
  • Technique: Check if segment passes through prism, like BH from top-back to bottom-front
  • Check: Verify endpoints are not connected by edges or face diagonals ✓

Common Mistakes

Avoid these frequent errors
  • Confusing face diagonals with space diagonals
    Don't identify segments like AC or HF as space diagonals = they're face diagonals! These lie on the surface faces of the prism, not through its interior. Always check that the diagonal passes completely through the prism's interior space.

Practice Quiz

Test your knowledge with interactive questions

A cuboid is shown below:

222333555

What is the surface area of the cuboid?

FAQ

Everything you need to know about this question

What's the difference between a face diagonal and a space diagonal?

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A face diagonal lies on one of the rectangular faces of the prism (like AC on the top face). A space diagonal passes through the interior from one vertex to the opposite vertex (like BH).

How do I identify opposite vertices in a rectangular prism?

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Opposite vertices are as far apart as possible - they don't share any edges or faces. In this prism, B (top-back-right) is opposite to H (bottom-front-left).

Why can't segments like HF be space diagonals?

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HF connects two vertices on the same vertical edge of the prism. A space diagonal must connect vertices that are diagonally opposite through the entire 3D space.

How many space diagonals does a rectangular prism have?

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A rectangular prism has exactly 4 space diagonals. Each connects a vertex to its unique opposite vertex through the interior space.

Is there a pattern to find space diagonals?

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Yes! Look for vertices that differ in all three dimensions: height (top/bottom), width (left/right), and depth (front/back). These pairs create space diagonals.

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