To solve this problem, we can use the Pythagorean theorem, which applies to right-angled triangles. Let's go through the steps one by one.
### Step 1: Understanding the problem
- We have a wall that is 12 meters high (let's label this as the vertical leg of the right-angled triangle, [tex]\(a = 12\)[/tex] meters).
- The base of the ladder is 2 meters away from the wall (let's label this as the horizontal leg of the triangle, [tex]\(b = 2\)[/tex] meters).
- We need to find out the length of the ladder, which will be the hypotenuse of the right-angled triangle formed by the wall, the ground, and the ladder.
### Step 2: Drawing the diagram
Below is a rough representation of what the setup looks like:
```
|\
| \
| \
| \
| \
| \
| \
| \
|________\
```
- The vertical line represents the wall with a height of 12 meters.
- The horizontal line at the bottom represents the distance from the base of the ladder to the wall, which is 2 meters.
- The inclined line represents the ladder, which we want to determine the length of.
### Step 3: Applying the Pythagorean theorem
The Pythagorean theorem states that in a right-angled triangle:
[tex]\[a^2 + b^2 = c^2\][/tex]
where
- [tex]\(a\)[/tex] is the length of one leg (wall height)
- [tex]\(b\)[/tex] is the length of the other leg (distance from the wall)
- [tex]\(c\)[/tex] is the length of the hypotenuse (ladder length)
### Step 4: Plugging in the values
Given:
- [tex]\(a = 12\)[/tex] meters
- [tex]\(b = 2\)[/tex] meters
We need to find [tex]\(c\)[/tex], the length of the ladder.
[tex]\[12^2 + 2^2 = c^2\][/tex]
### Step 5: Calculating the components
[tex]\[12^2 = 144\][/tex]
[tex]\[2^2 = 4\][/tex]
So,
[tex]\[144 + 4 = c^2\][/tex]
[tex]\[148 = c^2\][/tex]
### Step 6: Solving for [tex]\(c\)[/tex]
To find [tex]\(c\)[/tex], we take the square root of both sides:
[tex]\[c = \sqrt{148}\][/tex]
### Step 7: Finding the square root
[tex]\[c ≈ 12.165525060596439\][/tex]
Therefore, the length of the ladder is approximately 12.17 meters.
