ABCD is a square of side s. F is the midpoint of AB and E lies on BC such that BE = (1/3)·BC. If area of △FBE is 108 m^2, find the length of diagonal AC.

Introduction / Context:We work in a square and place points using simple fractions of the sides. Using coordinates or base–height reasoning, we can compute the area of △FBE in terms of the square side s, solve for s, and then compute the diagonal AC = s√2.

Given Data / Assumptions:

• ABCD is a square with side s.
• F is midpoint of AB ⇒ AF = FB = s/2.
• E lies on BC with BE = (1/3)·BC ⇒ E is one-third up from B toward C.
• Area(△FBE) = 108 m^2.

Concept / Approach:

• Set coordinates: A(0,0), B(s,0), C(s,s), D(0,s).
• Then F(s/2, 0) and E(s, s/3).
• Use area = (1/2)·base·height with base BF horizontal.

Step-by-Step Solution:

Verification / Alternative check:Coordinate shoelace formula with F(s/2,0), B(s,0), E(s,s/3) also yields area s^2/12, confirming the result.

Why Other Options Are Wrong:

• 63 m, 63√2 m, 72√2 m: Do not follow from s^2/12 = 108.
• None of these: Not applicable; 36√2 m is exact.

Common Pitfalls:

• Mistaking BE = s/3 for y(E) when coordinates are not carefully set.
• Using s/3 as base instead of height; here the base is BF = s/2.

Final Answer:36√2 m