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Kisachek [45]
3 years ago
15

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

ches tall?​

Mathematics
1 answer:
EastWind [94]3 years ago
5 0

Answer:

6\sqrt{19} \approx 26.153 inches.

Step-by-step 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} = \sqrt{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} = \sqrt{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(\sqrt{26^2 + 2^2}\right)^2 + 2^2}\\ &= \sqrt{\left(26^2 + 2^2\right) + 2^2} \\&= 6\sqrt{19} \\&\approx 26.153\end{aligned}.

Hence, the length of the longest line segment in this prism is 6\sqrt{19} \approx 26.153 inches.

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