Question

What is the measurement of the longest line segment in a right rectangular prism that is 26 inches long, 2 inches wide, and 2 inches tall?​

Answer 1

`6√(19) \approx 26.153` inches.

Explanation:

The longest line segment in a right rectangular prism is the diagonal that connects two opposite vertices. On the first diagram attached, the green line segment connecting A and G is one such diagonals. The goal is to find the length of segment
`\mathsf{AG}`.

In this diagram (not to scale,)
`\mathsf{AB} = 26` (length of prism,)
`\mathsf{AC} = 2` (width of prism,)
`\mathsf{AE} = 2` (height of prism.)

Pythagorean Theorem can help find the length of
`\mathsf{AG}`, one of the longest line segments in this prism. However, note that this theorem is intended for right triangles in 2D, not the diagonal in a 3D prism. The workaround is to simply apply this theorem on two different right triangles.

Start by finding the length of line segment
`\mathsf{AD}`. That's the black dotted line in the diagram. In right triangle
`\triangle\mathsf{ABD}` (second diagram,)

• Segment
  `\mathsf{AD}` is the hypotenuse.
• One of the legs of
  `\triangle\mathsf{ABD}` is
  `\mathsf{AB}`. The length of
  `\mathsf{AB}` is
  `26`, same as the length of this prism.
• Segment
  `\mathsf{BD}` is the other leg of this triangle. The length of
  `\mathsf{BD}` is
  `2`, same as the width of this prism.

Apply the Pythagorean Theorem to right triangle
`\triangle\mathsf{ABD}` to find the length of
`\mathsf{AB}`, the hypotenuse of this triangle:

`\mathsf{AD} = \sqrt{\mathsf{AB}^2 + \mathsf{BD}^2} = √(26^2 + 2^2)`.

Consider right triangle
`\triangle \mathsf{ADG}` (third diagram.) In this triangle,

• Segment
  `\mathsf{AG}` is the hypotenuse, while
• 
  `\mathsf{AD}` and
  `\mathsf{DG}` are the two legs.

`\mathsf{AD} = √(26^2 + 2^2)`. The length of segment
`\mathsf{DG}` is the same as the height of the rectangular prism,
`2` (inches.) Apply the Pythagorean Theorem to right triangle
`\triangle \mathsf{ADG}` to find the length of the hypotenuse
`\mathsf{AG}`:

`\begin{aligned}\mathsf{AG} &= \sqrt{\mathsf{AD}^2 + \mathsf{GD}^2} \\ &= \sqrt{\left(√(26^2 + 2^2)\right)^2 + 2^2}\\ &= √(\left(26^2 + 2^2\right) + 2^2) \\&= 6√(19) \\&\approx 26.153\end{aligned}`.

Hence, the length of the longest line segment in this prism is
`6√(19) \approx 26.153` inches.