What is the area of the given rectangular field? Statements: I. The perimeter…
2024
What is the area of the given rectangular field?
Statements:
I. The perimeter of the field is given.
II. The diagonal of the field is given.
- A.
Statement I alone is sufficient.
- B.
Statement II alone is sufficient.
- C.
Both the statements when put together are sufficient.
- D.
Both the statements put together are not sufficient.
Attempted by 2 students.
Show answer & explanation
Correct answer: C
CONCEPT: For a rectangle with length L and breadth B, the perimeter is P = 2(L + B), the diagonal D satisfies D² = L² + B² (Pythagoras), and the area A = L × B. These three are linked by one algebraic identity — (L + B)² = L² + B² + 2LB — which lets the product LB (the area) be isolated from P and D without ever solving for L and B individually.
APPLICATION:
From the perimeter (Statement I): L + B = P / 2.
From the diagonal (Statement II): L² + B² = D².
Square the first equation: (L + B)² = (P / 2)² = L² + B² + 2LB.
Substitute the diagonal relation for L² + B²: (P / 2)² = D² + 2LB, so 2LB = (P / 2)² − D², i.e. Area = LB = [(P / 2)² − D²] / 2.
Statement I alone fixes only L + B, and Statement II alone fixes only L² + B² — either one alone still allows many (L, B) pairs with different areas. Combined, the identity above turns the two separate facts into a single equation that outputs one, unique area.
CROSS-CHECK: Take L = 3, B = 4, so P = 14 and D = 5. The formula gives [(14 / 2)² − 5²] / 2 = (49 − 25) / 2 = 12, which matches the actual area 3 × 4 = 12 — confirming the identity.
Both statements together are sufficient to find the area; neither statement alone is sufficient.