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. The theorem 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 1: Identify the components of the triangle
- Let \( AB \) be the height of the ladder from the ground to the wall (the vertical side).
- Let \( BC \) be the distance from the foot of the ladder to the wall (the horizontal side), which is given as 5 m.
- Let \( AC \) be the length of the ladder (the hypotenuse), which is given as 13 m.

### Step 2: Write down the Pythagorean theorem
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]

### Step 3: Substitute the known values
We know:
- \( AC = 13 \) m
- \( BC = 5 \) m

Substituting these values into the equation:
\[ 13^2 = AB^2 + 5^2 \]

### Step 4: Calculate the squares
Calculating the squares:
\[ 169 = AB^2 + 25 \]

### Step 5: Rearrange the equation to solve for \( AB^2 \)
Subtract \( 25 \) from both sides:
\[ 169 - 25 = AB^2 \]
\[ 144 = AB^2 \]

### Step 6: Take the square root to find \( AB \)
Taking the square root of both sides:
\[ AB = \sqrt{144} \]
\[ AB = 12 \text{ m} \]

### Conclusion
The distance of the other end of the ladder from the ground is \( 12 \) meters.

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