Rectangle corner distances — find the minimum remaining distance given 3 m and 5 m (Age 13)

Step-by-step explanation for a 13-year-old: list all placements of 3 m and 5 m as sides or diagonal of a rectangle, reject impossible cases using diagonal ≥ side, use Pythagoras to solve for the unknown side, and conclude the minimum remaining distance is 4 m. Includes a 100-word Ally McBeal–style teacher comment.

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Place corner F at (0,0). Let the rectangle sides be a and b, and the diagonal be d = √(a^2 + b^2). The three distances from F are: a, b, and √(a^2 + b^2).

  1. List how 3 and 5 can fit into {a, b, √(a^2 + b^2)} — there are three natural assignments to test:
    1. a = 3, b = 5 (so the remaining distance is the diagonal d)
    2. a = 3, d = 5 (so b is unknown — use Pythagoras to find b)
    3. a = 5, d = 3 (test this but it will be impossible because diagonal ≥ each side)
  2. Work each case with Pythagoras and show algebra:
    • Case 1: a = 3, b = 5 ⇒ d = √(3^2 + 5^2) = √(9 + 25) = √34 ≈ 5.83 m.
    • Case 2: a = 3, d = 5 ⇒ √(3^2 + b^2) = 5. Square both sides: 9 + b^2 = 25 ⇒ b^2 = 16 ⇒ b = 4 (take positive root). So the remaining distance is 4 m.
    • Case 3: a = 5, d = 3 ⇒ impossible because a side (5 m) cannot be larger than the diagonal (3 m). Diagonals of rectangles are always at least as long as each side, so reject this case.
  3. Compare the valid remaining distances: 4 m and √34 m (≈ 5.83 m). The minimum is 4 m.

One-sentence justification: The smallest possible remaining distance is 4 m, because when 3 m is a side and 5 m is the diagonal Pythagoras gives the other side as 4 m, and the alternative valid configuration gives a larger diagonal √34 m while the remaining assignment is impossible since a diagonal cannot be smaller than a side.

Diagram note: Draw rectangle with F at (0,0), label adjacent sides a and b, mark right angles, and label diagonal d = √(a^2 + b^2). Place the given numbers 3 and 5 into the three tested slots as above.

Shortcut note: If you spot 3–4–5, call it out — here Case 2 is exactly the 3–4–5 right triangle so you get b = 4 instantly.

You performed like a soloist in a courthouse musical: theatrical, precise, and convincing. You fixed F at the origin, labelled sides a and b and diagonal d, and tried each assignment as if dressing characters for a play. You listed every case, rejected the impossible one with the short rule diagonal ≥ side, and showed tidy algebra: 9 + b^2 = 25 ⇒ b^2 = 16 ⇒ b = 4. Units and right angles were labelled. You noticed the 3-4-5 shortcut when it was pointed out. Keep revealing every step; your clear, dramatic reasoning earns full marks and applause always.

Final answer (compact): 4

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