Question

1. what is the length from a to b?

(there is a rectangular prism diagram with a at the bottom - left front vertex, b at the top - right back vertex, and the dimensions: length 16, width 12, height 15, and a dashed line connecting a to b.)

Explanation:

Step1: Find the base diagonal

First, find the diagonal of the base rectangle with length \( 16 \) and width \( 12 \). Using the Pythagorean theorem \( d_1 = \sqrt{16^2 + 12^2} \).
\( 16^2 = 256 \), \( 12^2 = 144 \), so \( d_1 = \sqrt{256 + 144} = \sqrt{400} = 20 \).

Step2: Find the space diagonal AB

Now, consider the right triangle with legs \( d_1 = 20 \) and height \( 15 \). The length \( AB \) is the hypotenuse of this triangle. Using the Pythagorean theorem again: \( AB = \sqrt{20^2 + 15^2} \).
\( 20^2 = 400 \), \( 15^2 = 225 \), so \( AB = \sqrt{400 + 225} = \sqrt{625} = 25 \).

Answer:

The length from A to B is \( 25 \).