A ladder 10m long rests against a vertical wall. If the foot of the ladder is 6m away from the wall and the ladder just reaches the top of the wall, how high is the wall?

Answer

To solve the problem, we will use the Pythagorean theorem, which states that in a right-angled 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 components of the triangle**:
- The ladder acts as the hypotenuse (AC) of the right triangle.
- The distance from the foot of the ladder to the wall (BC) is the base.
- The height of the wall (AB) is the perpendicular.

2. **Assign values**:
- Length of the ladder (AC) = 10 m (hypotenuse)
- Distance from the wall (BC) = 6 m (base)
- Height of the wall (AB) = ? (perpendicular)

3. **Apply the Pythagorean theorem**:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ 10^2 = AB^2 + 6^2 \]

4. **Calculate the squares**:
\[ 100 = AB^2 + 36 \]

5. **Rearrange the equation to find AB^2**:
\[ AB^2 = 100 - 36 \]
\[ AB^2 = 64 \]

6. **Take the square root to find AB**:
\[ AB = \sqrt{64} \]
\[ AB = 8 \text{ m} \]

7. **Conclusion**:
The height of the wall is 8 meters.