To find the diagonal length of the base rectangle of the rectangular prism, we will use the Pythagorean Theorem, which is \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse (the diagonal in this case) and \(a\) and \(b\) are the other two sides.
- Finding the diagonal of the base rectangle:
- Length (\(l\)) = 4
- Width (\(w\)) = 3
Using the Pythagorean Theorem:
\[ d_{\text{base}}^2 = l^2 + w^2 \] \[ d_{\text{base}}^2 = 4^2 + 3^2 = 16 + 9 = 25 \] \[ d_{\text{base}} = \sqrt{25} = 5 \]
The diagonal length of the base rectangle is 5.
- Finding the diagonal of the prism:
- Height (\(h\)) = 12
- The diagonal of the base (\(d_{\text{base}}\)) = 5
Now, we can use the Pythagorean Theorem again to find the diagonal of the prism:
\[ d_{\text{prism}}^2 = d_{\text{base}}^2 + h^2 \] \[ d_{\text{prism}}^2 = 5^2 + 12^2 \] \[ d_{\text{prism}}^2 = 25 + 144 = 169 \] \[ d_{\text{prism}} = \sqrt{169} = 13 \]
The length of the diagonal of the prism is 13.