To solve the problem step by step, we will analyze the distances given in both columns A and B and use the Pythagorean theorem to find the required distances.
### Step-by-Step Solution:
**Step 1: Analyze Column A**
- We need to find the distance between point A and point X, where point X is located 8 miles east of point P, and point P is located 6 miles north of point A.
**Step 2: Set up the points for Column A**
- Let’s place point A at the origin (0, 0).
- Point P will then be at (0, 6) since it is 6 miles north of A.
- Point X, being 8 miles east of point P, will be at (8, 6).
**Step 3: Calculate the distance AX using the Pythagorean theorem**
- The distance AX can be calculated as follows:
\[
AX = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Here, \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (8, 6) \).
\[
AX = \sqrt{(8 - 0)^2 + (6 - 0)^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ miles}
\]
**Step 4: Analyze Column B**
- We need to find the distance between point B and point Y, where point Y is located 6 miles west of point Q, and point Q is located 8 miles south of point B.
**Step 5: Set up the points for Column B**
- Let’s place point B at the origin (0, 0).
- Point Q will then be at (0, -8) since it is 8 miles south of B.
- Point Y, being 6 miles west of point Q, will be at (-6, -8).
**Step 6: Calculate the distance BY using the Pythagorean theorem**
- The distance BY can be calculated as follows:
\[
BY = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Here, \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (-6, -8) \).
\[
BY = \sqrt{(-6 - 0)^2 + (-8 - 0)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ miles}
\]
**Step 7: Compare the distances**
- From our calculations, we found that:
- Distance AX = 10 miles
- Distance BY = 10 miles
Since both distances are equal, we conclude that:
**Final Answer:**
- The correct option is C: The columns are equal.
