A ladder 13 m long just reaches a window that is 12 m above the ground on one side of a street. Keeping its foot fixed at the same point on the ground, the ladder is then turned to the other side of the street and now just reaches a window 5 m high. What is the width (in metres) of the street?

Introduction:
This problem uses the Pythagoras theorem in two different right-angled triangles formed by a ladder leaning against buildings on opposite sides of a street. Because the ladder length remains fixed, and only the vertical heights change, we can compute the horizontal distances on each side and add them to get the street's width.

Given Data / Assumptions:

• Length of the ladder (hypotenuse) = 13 m.
• Height of the window on one side = 12 m.
• Height of the window on the opposite side = 5 m.
• The ladder foot stays at the same point on the ground between the two buildings.
• The ground is level and the buildings are vertical, forming right-angled triangles with the ladder.

Concept / Approach:
On each side, the ladder, the building height, and the ground form a right-angled triangle. We can find the horizontal distance from the ladder foot to each building using:
(horizontal distance)² + (height)² = (ladder length)².The width of the street is the sum of the horizontal distances to the two buildings.

Step-by-Step Solution:
Step 1: First side (12 m window): Let x₁ be the horizontal distance.Step 2: Apply Pythagoras: x₁² + 12² = 13² ⇒ x₁² + 144 = 169 ⇒ x₁² = 25 ⇒ x₁ = 5 m.Step 3: Second side (5 m window): Let x₂ be the horizontal distance.Step 4: Apply Pythagoras: x₂² + 5² = 13² ⇒ x₂² + 25 = 169 ⇒ x₂² = 144 ⇒ x₂ = 12 m.Step 5: Width of the street = x₁ + x₂ = 5 + 12 = 17 m.

Verification / Alternative check:
You can verify that the pairs (5,12,13) and (12,5,13) each satisfy the Pythagoras theorem: 5² + 12² = 25 + 144 = 169 = 13². This confirms that the horizontal distances are calculated correctly and that adding them gives the true street width.

Why Other Options Are Wrong:
Options like 14 m, 15 m, 16 m, or 19 m arise if you miscompute one of the horizontal distances or incorrectly subtract instead of adding them. None of these satisfy the two right triangles with the same 13 m ladder and the given heights of 12 m and 5 m.

Common Pitfalls:
Students sometimes think the street width is the difference, not the sum, of the two horizontal distances, or they swap the roles of heights and hypotenuse in the Pythagoras formula. Always remember that the ladder is the hypotenuse and that the buildings lie on opposite sides of the fixed foot of the ladder, so distances must be added.

Final Answer:
The width of the street is 17 m.