Dear student (age 13),
Short mathematical answer (with units): F to A = 4 m.
Why, step by step:
- Label the rectangle so that the two side lengths are x and y, and the diagonal is d = sqrt(x^2 + y^2).
- The three distances from F to the other corners must be {x, y, d} in some order. We are given two of them: 3 and 5.
- Consider the consistent possibilities:
- Case 1: The two given numbers are the two side lengths, x = 3 and y = 5. Then the remaining distance (the diagonal) is d = sqrt(3^2 + 5^2) = sqrt(34) ≈ 5.83 m.
- Case 2: One given number is a side and the other is the diagonal. The only consistent way is side = 3 and diagonal = 5 (the alternative, side = 5 and diagonal = 3, is impossible because a diagonal is longer than each side). Then the other side is y = sqrt(d^2 - x^2) = sqrt(5^2 - 3^2) = sqrt(25 - 9) = 4 m.
- Compare the missing distances from the two cases: sqrt(34) ≈ 5.83 m (Case 1) and 4 m (Case 2). The minimum possible missing distance is 4 m, so F–A = 4 m is the least possible.
Feedback in Ally McBeal cadence: You delivered the correct number — a crisp, confident “4” — and your spoken explanation showed you really understand Pythagoras. But Alcumus wants the paper trail: draw or describe the rectangle, label sides and diagonal, write the Pythagorean equation, and always include units (m). One neat line of algebra beats mystery: define variables, write d^2 = x^2 + y^2, solve, and conclude with “4 m.”
Future targets (next year, AoPS Introduction to Algebra): practice short written proofs and clear justifications. Goals: always label diagrams (or describe them), define variables, state equations used, consider cases explicitly, and finish with units. Work on writing two- or three-line proofs that show why each case is allowed or impossible. This will turn your strong verbal reasoning into rock-solid written solutions.
Well done — correct concept, correct number; next time show the work and unit and you’ll earn full marks every time.