The upper end of a ladder touches a 12 meter high window on one side of a street. The ladder is rotated to opposite side of the street keeping its foot fixed and touches a 9 meter high window. If length of the ladder is 15 m, then find the width of the road.

Answer

To solve the problem, we will use the Pythagorean theorem, which 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-by-Step Solution:

1. **Understand the Setup**:
- We have a ladder of length 15 meters.
- When the ladder touches a window 12 meters high on one side of the street, we can form a right triangle.
- When the ladder is rotated to touch a window 9 meters high on the opposite side, we form another right triangle.

2. **Label the Triangles**:
- Let the foot of the ladder be point C.
- Let the point where the ladder touches the 12-meter window be point A.
- Let the point where the ladder touches the 9-meter window be point D.
- The height of the window on one side is 12 meters (point A) and on the other side is 9 meters (point D).
- Let the width of the road be represented as BC (the distance from point B to point C).

3. **Apply the Pythagorean Theorem for Triangle ABC**:
- In triangle ABC, we have:
\[ AB^2 = AC^2 + BC^2 \]
- Here, \(AB = 15\) m (length of the ladder), \(AC = 12\) m (height of the window).
- Plugging in the values:
\[ 15^2 = 12^2 + BC^2 \]
\[ 225 = 144 + BC^2 \]
- Rearranging gives:
\[ BC^2 = 225 - 144 = 81 \]
- Taking the square root:
\[ BC = \sqrt{81} = 9 \text{ m} \]

4. **Apply the Pythagorean Theorem for Triangle ADC**:
- In triangle ADC, we have:
\[ AD^2 = AC^2 + CD^2 \]
- Here, \(AD = 15\) m (length of the ladder), \(AC = 9\) m (height of the window).
- Plugging in the values:
\[ 15^2 = 9^2 + CD^2 \]
\[ 225 = 81 + CD^2 \]
- Rearranging gives:
\[ CD^2 = 225 - 81 = 144 \]
- Taking the square root:
\[ CD = \sqrt{144} = 12 \text{ m} \]

5. **Find the Width of the Road**:
- The total width of the road is the sum of \(BC\) and \(CD\):
\[ \text{Width of the road} = BC + CD = 9 + 12 = 21 \text{ m} \]

### Final Answer:
The width of the road is **21 meters**.