A `10` m long ladder is placed against a wall in such a way that the foot of the ladder is `6` m away from the wall. Up to what height does the ladder reach the wall ?

Options

A. `5m`
B. `6m`
C. `8m`
D. `9m`

Answer

To solve the problem of how high the ladder reaches the wall, we can use the Pythagorean theorem. Here's a step-by-step solution:

### Step 1: Understand the Problem
We have a right triangle formed by the ladder, the wall, and the ground. The ladder acts as the hypotenuse, the height at which the ladder touches the wall is one leg, and the distance from the wall to the foot of the ladder is the other leg.

### Step 2: Identify the Lengths
- Length of the ladder (hypotenuse, AB) = 10 m
- Distance from the wall to the foot of the ladder (BC) = 6 m
- Height reached on the wall (AC) = x (this is what we need to find)

### Step 3: Apply the Pythagorean Theorem
According to the Pythagorean theorem:
\[ AB^2 = AC^2 + BC^2 \]

Substituting the known values:
\[ 10^2 = x^2 + 6^2 \]

### Step 4: Calculate the Squares
Calculating the squares:
\[ 100 = x^2 + 36 \]

### Step 5: Rearrange the Equation
Now, rearranging the equation to isolate \( x^2 \):
\[ x^2 = 100 - 36 \]

### Step 6: Perform the Subtraction
Calculating the right side:
\[ x^2 = 64 \]

### Step 7: Take the Square Root
Now, take the square root of both sides to find \( x \):
\[ x = \sqrt{64} \]

### Step 8: Determine the Height
Calculating the square root:
\[ x = 8 \]

Thus, the height at which the ladder reaches the wall is **8 meters**.

### Final Answer
The ladder reaches a height of **8 meters** on the wall.

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