A ladder is resting against a vertical wall, and initially the bottom of the ladder is 2.5 m away from the wall along the ground. If the ladder slips so that its top slides 0.8 m down the wall and, at the same time, the bottom moves 1.4 m farther away from the wall, what is the fixed length (in metres) of the ladder?

Introduction:
This problem uses right-angled triangles and the Pythagoras theorem to determine the fixed length of a ladder as its position changes. The ladder slides while maintaining the same length, giving us two different right triangles with the same hypotenuse. By equating the two expressions for the ladder length, we can solve for its value.

Given Data / Assumptions:

• Initially, distance from wall to foot of ladder along the ground = 2.5 m.
• The top of the ladder is at some height h above the ground.
• The ladder then slips 0.8 m down, so new height = h − 0.8 m.
• The bottom moves 1.4 m away, so new horizontal distance = 2.5 + 1.4 = 3.9 m.
• The ladder length remains constant throughout.

Concept / Approach:
We model both situations as right-angled triangles. If L is the ladder length, then:
L² = 2.5² + h² (initial position)L² = 3.9² + (h − 0.8)² (after slipping)Since L² is the same, we equate these two expressions and solve for h, then back-calculate L.

Step-by-Step Solution:
Step 1: Write the first equation: L² = 2.5² + h² = 6.25 + h².Step 2: Write the second equation: L² = 3.9² + (h − 0.8)² = 15.21 + h² − 1.6h + 0.64.Step 3: Simplify the second: L² = 15.85 + h² − 1.6h.Step 4: Set the two expressions equal: 6.25 + h² = 15.85 + h² − 1.6h.Step 5: Cancel h² from both sides, giving 6.25 = 15.85 − 1.6h.Step 6: Rearrange: −9.6 = −1.6h, so h = 9.6 / 1.6 = 6 m.Step 7: Use the initial triangle to find L: L² = 6.25 + 6² = 6.25 + 36 = 42.25.Step 8: Therefore, L = √42.25 = 6.5 m.

Verification / Alternative check:
Check with the second position: L² should also equal 3.9² + (6 − 0.8)² = 15.21 + 5.2² = 15.21 + 27.04 = 42.25, giving L = 6.5 m again. This confirms consistency.

Why Other Options Are Wrong:
Values like 6.2 m, 6.8 m, 7.5 m, or 5.9 m do not satisfy both right-triangle configurations simultaneously. When substituted, they fail to give the same L for both positions, violating the fixed ladder-length condition.

Common Pitfalls:
Common errors include forgetting that the ladder length is constant, mishandling the algebra when expanding (h − 0.8)², or rounding too early. Carefully expanding the square and equating both expressions for L² ensures an accurate answer.

Final Answer:
The length of the ladder is 6.5 m.