Question

there is a rectangular prism. height, 12. length, 4. width, 3. First, we will use side lengths 3 and 4 in the Pythagorean Theorem to find the diagonal length of the base rectangle. Second, we will use 12 and the length we found in step one to solve for the diagonal of the prism. The length of the diagonal of the prism is ___________

Answer 1

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.

1. **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.

2. **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**.