To find the longest line segment that can be drawn in a right rectangular prism (also known as a rectangular box), we need to calculate the length of the space diagonal. The formula to find the space diagonal \(d\) of a rectangular prism with length \(l\), width \(w\), and height \(h\) is given by:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
In this case, we have:
- Length \(l = 13 , \text{cm}\)
- Width \(w = 8 , \text{cm}\)
- Height \(h = 7 , \text{cm}\)
Now we can substitute these values into the formula:
\[ d = \sqrt{(13)^2 + (8)^2 + (7)^2} \]
Calculating the squares:
\[ d = \sqrt{169 + 64 + 49} \]
Adding these values together:
\[ d = \sqrt{282} \]
Now calculating the square root:
\[ d \approx 16.82 , \text{cm} \]
Therefore, the longest line segment that can be drawn in the rectangular prism is approximately \(16.82 , \text{cm}\).