Rectangles – Length is twice breadth, area change after adjusting sides:\nA rectangle has length equal to twice its breadth. If the length is decreased by 5 cm and the breadth is increased by 5 cm, the area increases by 75 sq cm. Find the original length.

Introduction / Context:
Area of a rectangle depends on length and breadth. Linear changes to sides produce a quadratic change in area, solvable by forming and solving an equation.

Given Data / Assumptions:

• Let breadth = b cm; length = 2b cm.
• New dimensions: (2b − 5) and (b + 5).
• Area increase = 75 sq cm.

Concept / Approach:
Set up: (2b − 5)(b + 5) − (2b)(b) = 75 and solve for b, then compute original length 2b.

Step-by-Step Solution:

Verification / Alternative check:
Check areas: Original = 2b^2 = 800. New = (35)(25) = 875 ⇒ increase = 75 ✔️

Why Other Options Are Wrong:
10, 20, 30 cm refer to breadths or miscomputed lengths; only 40 cm matches the derived length. From the options, 20 cm corresponds to breadth, not length; thus the correct length is 40 cm. (Since choices list 40 cm, select that.)

Common Pitfalls:
Sign mistakes while expanding; mixing up which dimension is doubled; forgetting area increase applies to full rectangle.

Final Answer:
20 cm (Note: Option '40 cm' should be chosen for length; if the interface expects the length value, pick 40 cm.)