A ladder `17` m long reaches a window which is `8` 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 at a height of `15` m. Find the width of the street.

Answer

To solve the problem step by step, we will use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

### Step 1: Understand the Problem
We have a ladder of length 17 m that reaches a window 8 m above the ground on one side of the street. We need to find the width of the street when the ladder is turned to reach another window 15 m above the ground on the opposite side.

### Step 2: Draw a Diagram
Draw a diagram to visualize the scenario:
- Let A be the point where the ladder touches the ground.
- Let B be the point where the ladder touches the first window (8 m high).
- Let C be the point where the ladder touches the second window (15 m high).
- Let O be the foot of the ladder (point A).
- Let AO be the distance from the foot of the ladder to the wall with the first window.
- Let OC be the distance from the foot of the ladder to the wall with the second window.

### Step 3: Apply Pythagorean Theorem for the First Window
For the first window (height = 8 m):
- The ladder (hypotenuse) = 17 m
- Height (one side) = 8 m
- Distance from the wall (other side) = AO

Using the Pythagorean theorem:
\[ BO^2 = AB^2 + AO^2 \]
\[ 17^2 = 8^2 + AO^2 \]

### Step 4: Calculate AO
Calculating the squares:
\[ 289 = 64 + AO^2 \]
\[ AO^2 = 289 - 64 \]
\[ AO^2 = 225 \]
Taking the square root:
\[ AO = \sqrt{225} = 15 \text{ m} \]

### Step 5: Apply Pythagorean Theorem for the Second Window
For the second window (height = 15 m):
- The ladder (hypotenuse) = 17 m
- Height (one side) = 15 m
- Distance from the wall (other side) = OC

Using the Pythagorean theorem:
\[ DO^2 = DC^2 + OC^2 \]
\[ 17^2 = 15^2 + OC^2 \]

### Step 6: Calculate OC
Calculating the squares:
\[ 289 = 225 + OC^2 \]
\[ OC^2 = 289 - 225 \]
\[ OC^2 = 64 \]
Taking the square root:
\[ OC = \sqrt{64} = 8 \text{ m} \]

### Step 7: Find the Width of the Street
The total width of the street is the sum of AO and OC:
\[ \text{Width of the street} = AO + OC \]
\[ \text{Width of the street} = 15 + 8 = 23 \text{ m} \]

### Final Answer
The width of the street is **23 meters**.

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