A 6.5 m long ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall. Find the height of the wall where the top of the ladder touches it

Answer

To solve the problem of finding the height of the wall where the top of the ladder touches it, we can use the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Identify the components
We have a right triangle formed by the ladder, the wall, and the ground. Let's denote:
- AC = Length of the ladder = 6.5 m
- BC = Distance from the wall to the foot of the ladder = 2.5 m
- AB = Height of the wall where the ladder touches = ?

### Step 2: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[ AB^2 + BC^2 = AC^2 \]
Substituting the known values:
\[ AB^2 + (2.5)^2 = (6.5)^2 \]

### Step 3: Calculate the squares
Calculate \( (2.5)^2 \) and \( (6.5)^2 \):
- \( (2.5)^2 = 6.25 \)
- \( (6.5)^2 = 42.25 \)

### Step 4: Set up the equation
Now we can write the equation:
\[ AB^2 + 6.25 = 42.25 \]

### Step 5: Solve for \( AB^2 \)
Subtract 6.25 from both sides:
\[ AB^2 = 42.25 - 6.25 \]
\[ AB^2 = 36 \]

### Step 6: Find \( AB \)
Now take the square root of both sides to find \( AB \):
\[ AB = \sqrt{36} \]
\[ AB = 6 \text{ m} \]

### Conclusion
The height of the wall where the top of the ladder touches is 6 meters.

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