Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using the above, solve the following: A ladder reaches a window which is 12 m above the ground on one side of the street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window 9m high. Find the width of the street if the length of the ladder is 15 m

Answer

To solve the problem step by step, we will first prove the Pythagorean theorem and then use it to find the width of the street.

### Step 1: Prove the Pythagorean Theorem

1. **Construct a Right Triangle**: Let triangle ABC be a right triangle where angle B is the right angle. The sides opposite angles A and C are denoted as a and b, respectively, and the hypotenuse (the side opposite the right angle) is denoted as c.

2. **Draw a Perpendicular**: Draw a perpendicular line from point B to side AC, meeting AC at point D. This creates two smaller triangles: triangle ABD and triangle BCD.

3. **Establish Similarity**: By the AA (Angle-Angle) similarity criterion, triangle ABD is similar to triangle ABC, and triangle BCD is also similar to triangle ABC.

4. **Set Up Proportions**: From the similarity of triangles:
- For triangle ABD:
\[ \frac{AD}{AB} = \frac{AB}{AC} \implies AB^2 = AD \cdot AC \quad \text{(1)} \]
- For triangle BCD:
\[ \frac{CD}{BC} = \frac{BC}{AC} \implies BC^2 = CD \cdot AC \quad \text{(2)} \]

5. **Add the Equations**: Adding equations (1) and (2):
\[ AB^2 + BC^2 = AD \cdot AC + CD \cdot AC \]
\[ AB^2 + BC^2 = AC(AD + CD) \]

6. **Simplify**: Since \(AD + CD = AC\), we have:
\[ AB^2 + BC^2 = AC^2 \]

7. **Conclusion**: Thus, we have proved that in a right triangle, the square of the hypotenuse (AC) is equal to the sum of the squares of the other two sides (AB and BC).

### Step 2: Solve the Ladder Problem

1. **Identify the Given Information**:
- Length of the ladder (hypotenuse, c) = 15 m
- Height of the first window (height from the ground, b) = 12 m
- Height of the second window (height from the ground, a) = 9 m

2. **Set Up the Right Triangle for the First Window**:
- Let BC be the distance from the foot of the ladder to the wall where the first window is located.
- Using the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \implies 15^2 = BC^2 + 12^2 \]
\[ 225 = BC^2 + 144 \]
\[ BC^2 = 225 - 144 = 81 \]
\[ BC = \sqrt{81} = 9 \text{ m} \]

3. **Set Up the Right Triangle for the Second Window**:
- Let AE be the distance from the foot of the ladder to the wall where the second window is located.
- Using the Pythagorean theorem again:
\[ c^2 = a^2 + b^2 \implies 15^2 = AE^2 + 9^2 \]
\[ 225 = AE^2 + 81 \]
\[ AE^2 = 225 - 81 = 144 \]
\[ AE = \sqrt{144} = 12 \text{ m} \]

4. **Calculate the Width of the Street**:
- The total width of the street (AB + BC) is:
\[ \text{Width of the street} = BC + AE = 9 + 12 = 21 \text{ m} \]

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