A 25 m long ladder is resting against a vertical wall. Initially, the foot of the ladder is 7 m away from the wall. If the top of the ladder then slides down the wall by 4 m, how far will its foot move away from the wall (in metres)?

Introduction / Context:
This problem combines the Pythagorean theorem with a dynamic situation: a ladder sliding down a wall. The ladder length remains constant, but its orientation and contact points change. We are asked to find how much the base of the ladder moves when the top descends by a known vertical distance.

Given Data / Assumptions:

• Length of the ladder (hypotenuse) = 25 m.
• Initial horizontal distance from the wall (base) = 7 m.
• The wall is vertical and the ground is horizontal.
• The top of the ladder slides down by 4 m along the wall.
• The ladder does not slip at the wall or ground and remains straight.

Concept / Approach:
The ladder, wall and ground form a right triangle. The ladder is the hypotenuse, the vertical distance up the wall is one leg, and the horizontal distance from the wall is the other leg. Using the Pythagorean theorem both before and after the movement, we determine the initial and final horizontal distances. The difference between these two distances is the required movement of the foot.

Step-by-Step Solution:
Ladder length L = 25 m.Initial base distance b1 = 7 m.Initial height h1 satisfies: L^2 = h1^2 + b1^2.25^2 = h1^2 + 7^2 ⇒ 625 = h1^2 + 49.h1^2 = 625 − 49 = 576 ⇒ h1 = 24 m.The top slides down 4 m, so new height h2 = 24 − 4 = 20 m.Again, L^2 = h2^2 + b2^2.625 = 20^2 + b2^2 = 400 + b2^2.b2^2 = 625 − 400 = 225 ⇒ b2 = 15 m.Movement of the foot = b2 − b1 = 15 − 7 = 8 m.

Verification / Alternative check:
We can check quickly by verifying both triangles: (24, 7, 25) and (20, 15, 25) are Pythagorean triples since 24^2 + 7^2 = 576 + 49 = 625 and 20^2 + 15^2 = 400 + 225 = 625, both equal to 25^2. So the geometry is consistent and the base movement of 8 m is correct.

Why Other Options Are Wrong:
6 m, 9 m, 10 m, and 5 m: All are arbitrary shifts that do not arise from correct Pythagorean calculations; they result from errors like subtracting heights directly or miscomputing the new base distance.

Common Pitfalls:
Students sometimes assume that the difference in heights equals the difference in base distances, which is incorrect. Others forget that the ladder length is constant and fail to apply the Pythagorean theorem properly a second time. Squaring errors (e.g., miscomputing 25^2 or 20^2) also commonly lead to incorrect results.

Final Answer:
The foot of the ladder moves away from the wall by 8 m.