This problem involves determining the horizontal distance from the base of the stairs to the taller column.

Step 1: Understand the Problem

From the diagram:

• The stairs create a right triangle with:
  • The hypotenuse of 12 feet.
  • One vertical leg of 6 feet.
  • One horizontal leg is the total horizontal distance from the base to the taller column, which includes two segments:
    • 4 feet (to the shorter column).
    • The distance to the taller column, which we need to find.

Step 2: Use the Pythagorean Theorem

The Pythagorean theorem states: $\text{(Hypotenuse)}^2 = \text{(Horizontal leg)}^2 + \text{(Vertical leg)}^2$ Substitute the known values: $12^2 = (\text{Total horizontal distance})^2 + 6^2$ Simplify: $144 = (\text{Total horizontal distance})^2 + 36$ $(\text{Total horizontal distance})^2 = 144 - 36 = 108$ $\text{Total horizontal distance} = \sqrt{108} \approx 10.39 \, \text{feet}$

Step 3: Solve for the Unknown Distance

The total horizontal distance is 10.39 feet. Since 4 feet are already accounted for, the distance to the taller column is: $10.39 - 4 \approx 6.39 \, \text{feet.}$

Final Answer:

The distance from the base of the stairs to the taller column is approximately 6.39 feet.

Would you like a breakdown of these calculations or have any related questions?

5 Related Questions:

1. How is the Pythagorean theorem used to find unknown distances in triangles?
2. What are some practical examples of right triangle problems like this one?
3. Can the solution be verified by measuring angles?
4. How does changing the dimensions of the triangle affect the solution?
5. What are some tips for estimating square roots in real-life calculations?

Tip:

When solving problems involving right triangles, always double-check which side corresponds to the hypotenuse, as it is critical for using the Pythagorean theorem correctly.