Let's solve this problem using the Pythagorean Theorem, which is applicable to right-angled triangles.

Let:

• $L$ be the length of the ladder.
• The height from the ground to the top of the ladder (the vertical side of the triangle) is $L - 4$ feet, since the height is 4 feet less than the length of the ladder.
• The distance from the bottom of the ladder to the building (the horizontal side of the triangle) is 8 feet.

According to the Pythagorean Theorem:

$\text{(height)}^2 + \text{(base)}^2 = \text{(hypotenuse)}^2$

Substitute the values:

$(L - 4)^2 + 8^2 = L^2$

Step 1: Expand and simplify the equation

First, expand both sides:

$(L - 4)^2 = L^2 - 8L + 16$

$8^2 = 64$

So, the equation becomes:

$L^2 - 8L + 16 + 64 = L^2$

Simplify further:

$L^2 - 8L + 80 = L^2$

Step 2: Cancel out $L^2$ from both sides

$-8L + 80 = 0$

Step 3: Solve for $L$

$-8L = -80$

$L = 10$

Final Answer:

The length of the ladder is 10 feet.

Would you like further details or explanations on any part of this problem?

Here are some related questions:

1. How would you solve similar problems where the ladder is leaning at a different angle?
2. What would change if the distance from the bottom of the ladder to the building was different?
3. How does the Pythagorean theorem apply to 3-dimensional objects?
4. Can the ladder form any other angles with the ground and still satisfy the condition?
5. How can this problem be generalized to a different type of triangle?

Tip: Always double-check the setup of your triangle before applying the Pythagorean theorem to make sure you are using the correct sides for the equation.