Rectangular Prism Diagonals: Identifying Equal Interior Lines in 3D Geometry

Space Diagonals with Equal Length Analysis

Look at the dotted diagonal in the figure below.

Which diagonals of the rectangular prism are equal?

AAABBBCCCDDDEEEFFFGGGHHH

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:06 How many matching diagonals are there with the one we've drawn? Let's find out!
00:11 Imagine the face of a box. Each face is a rectangle.
00:16 In a rectangle, both diagonals are exactly the same length.
00:21 A box has opposite faces that are identical.
00:31 So, equal rectangles mean all their diagonals are equal too!
00:42 And that's how we solve this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the dotted diagonal in the figure below.

Which diagonals of the rectangular prism are equal?

AAABBBCCCDDDEEEFFFGGGHHH

2

Step-by-step solution

Let's look at the face AEHD

In it, there is another diagonal equal to ED, which is AH

Let's look at the face BFGC which is identical and equal to AEHD, in which there are two diagonals that are also equal to ED:

FC=BG=ED FC=BG=ED

3

Final Answer

AH,BG,FC AH,BG,FC

Key Points to Remember

Essential concepts to master this topic
  • Space Diagonals: Connect opposite vertices through rectangular prism interior
  • Equal Diagonals: Face diagonals AH, BG, FC all cross from vertex to opposite vertex
  • Verification: Check each diagonal connects vertices separated by all three dimensions ✓

Common Mistakes

Avoid these frequent errors
  • Confusing face diagonals with space diagonals
    Don't identify edge lines or face diagonals as space diagonals = wrong diagonal type! Face diagonals lie within rectangular faces, while space diagonals cross through the interior. Always identify diagonals that connect vertices separated by length, width, AND height.

Practice Quiz

Test your knowledge with interactive questions

Below is a cuboid with a length of

8 cm.

Its width is 2 cm and its height is

4 cm.

Calculate the volume of the cube.

222888444

FAQ

Everything you need to know about this question

What makes diagonals equal in a rectangular prism?

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All space diagonals are equal because they span the same three dimensions. Each diagonal like AH, BG, and FC connects vertices that are separated by the full length, width, and height of the prism.

How can I tell if a line is a space diagonal?

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A space diagonal must connect two vertices that are opposite corners of the rectangular prism. Look for lines that go through the interior, not along faces or edges.

Why isn't ED equal to the other diagonals shown?

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ED is a face diagonal that lies flat within one rectangular face. It's shorter than space diagonals because it only spans two dimensions (length and width) instead of all three.

Are there other equal diagonals not listed in the answer?

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Yes! A rectangular prism has 4 space diagonals total. Besides AH, BG, and FC, there's also diagonal EG. All four space diagonals have equal length.

How do I identify which vertices are opposite?

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Opposite vertices are separated by the maximum distance possible in the prism. If you can travel along three perpendicular edges to get from one vertex to another, they're opposite.

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