Question

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 what?

Answer 1

To find the diagonal length of the base rectangle with side lengths 3 and 4, we use the Pythagorean Theorem, which states:

\[
c = \sqrt{a^2 + b^2}
\]

where \( a \) and \( b \) are the lengths of the sides, and \( c \) is the length of the diagonal.

1. **Calculate the diagonal of the rectangle**:

Given \( a = 3 \) and \( b = 4 \):

\[
c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

So, the diagonal of the base rectangle is \( 5 \).

2. **Calculate the diagonal of the prism**:

The diagonal \( d \) of a rectangular prism can be found using the formula:

\[
d = \sqrt{l^2 + w^2 + h^2}
\]

In this case, the base rectangle is already calculated (with diagonal \( 5 \)) and we need to consider the height of the prism, which is given as \( h = 12 \). We can treat the diagonal of the base as one of the dimensions.

Thus, we can rewrite the formula incorporating the base diagonal:

\[
d = \sqrt{(diagonal\ of\ base)^2 + h^2}
\]

Substituting the values:

\[
d = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]

Therefore, the length of the diagonal of the prism is \( \boxed{13} \).