A ladder 13 m long rests against a vertical wall. If the foot of the ladder is 5 m from the foot of the wall, find the distance of the other end of the ladder from the ground.

Answer

To solve the problem step by step, we will use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

### Step-by-Step Solution:

1. **Identify the Triangle**:
- Let \( A \) be the point where the ladder touches the wall.
- Let \( B \) be the point where the foot of the ladder is placed on the ground.
- Let \( C \) be the point on the ground directly below point \( A \).
- Here, \( AC \) is the length of the ladder (hypotenuse), \( BC \) is the distance from the wall to the foot of the ladder (base), and \( AB \) is the height of the ladder on the wall (perpendicular).

2. **Assign the Values**:
- Length of the ladder \( AC = 13 \, \text{m} \)
- Distance from the wall \( BC = 5 \, \text{m} \)
- We need to find \( AB \).

3. **Apply the Pythagorean Theorem**:
- According to the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]

4. **Substitute the Known Values**:
- Substitute \( AC = 13 \, \text{m} \) and \( BC = 5 \, \text{m} \):
\[ AB^2 + 5^2 = 13^2 \]
\[ AB^2 + 25 = 169 \]

5. **Solve for \( AB^2 \)**:
- Rearrange the equation to isolate \( AB^2 \):
\[ AB^2 = 169 - 25 \]
\[ AB^2 = 144 \]

6. **Find \( AB \)**:
- Take the square root of both sides to find \( AB \):
\[ AB = \sqrt{144} = 12 \, \text{m} \]

### Final Answer:
The distance of the other end of the ladder from the ground is \( 12 \, \text{m} \).