Leaning against a wall is a ladder. The bottom of the ladder is 6 feet from the wall, while the top of the ladder is 10 feet above the ground. What is the ladder's length?

The problem involves a right triangle where the ladder forms the hypotenuse, the distance from the wall to the bottom of the ladder is one leg, and the height of the ladder on the wall is the other leg. To find the length of the ladder, we can use the **Pythagorean theorem**:

\[
a^2 + b^2 = c^2
\]

Where:
- \(a = 6\) feet (the distance from the wall),
- \(b = 10\) feet (the height of the ladder on the wall),
- \(c\) is the length of the ladder (the hypotenuse).

Substitute the values into the equation:

\[
6^2 + 10^2 = c^2
\]

\[
36 + 100 = c^2
\]

\[
c^2 = 136
\]

Now, take the square root of both sides:

\[
c = \sqrt{136} \approx 11.66 \text{ feet}
\]

Thus, the length of the ladder is approximately **11.66 feet**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the length of the ladder change if the distance from the wall was increased?
2. What happens if the height on the wall is less than the distance from the wall?
3. How does the Pythagorean theorem apply to real-world structures like ladders and buildings?
4. Can you find the angle at which the ladder meets the ground using trigonometry?
5. What is the area of the right triangle formed by the ladder, wall, and ground?

**Tip:** Whenever you have a right triangle problem, the Pythagorean theorem is a powerful tool for finding missing side lengths.

This problem demonstrates how to find the length of a ladder leaning against a wall using the Pythagorean theorem. Given the distance from the wall (6 feet) and the height of the ladder (10 feet), learn how to calculate the ladder's length, approximately 11.66 feet. Suitable for students in Grades 8-10.

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