A ladder is leaning against a building so that the distance from the ground to the top of the ladder is 4 feet less than the length of the ladder. Find the length of the ladder if the distance from the bottom of the ladder to the building is 8 feet.

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.

Solve this geometry problem using the Pythagorean theorem. A ladder is leaning against a building, and the distance from the top of the ladder to the ground is 4 feet less than the ladder's length. Learn how to apply the Pythagorean theorem to find the ladder's length with a step-by-step solution. Suitable for Grades 8-10.

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