13-year-old Alcumus drill: Rectangle corners — find minimum distance F–A (answer 4 m)

Clear, step-by-step explanation and a 150-word homeschool report for a 13-year-old. Shows why the minimum distance from F to A is 4 m using Pythagoras, lists missing submission items (diagram, units, written justification), and gives a model written solution to include on Alcumus.

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Short homeschool report (Ally McBeal cadence, 150 words):

Dear Parent, picture Ally McBeal narrating across the courtroom: your 13‑year‑old walked into a geometric scene, poised at corner F, answered — 4 m. They caught the right tune: recognizing that rectangle corners give two side lengths and a diagonal, and that 3 and 5 must be a side and a diagonal, so the missing side is 4 by Pythagoras. Bravo for conceptual clarity. A small procedural note: on Alcumus the student entered the correct final number but left out the units “m”, offered no written steps, and couldn’t attach a sketch. For homeschool reporting, please encourage brief written justification and a labelled rectangle: it’s how mathematical thinking becomes visible. Keep praising the strong reasoning — the work is right — and remind them next time to show a diagram, write units, and hand in one justification line so the answer sings proudly on paper as well as in conversation.

Step-by-step math explanation (what to write next time):

  1. Label the rectangle so F is one corner; let the two adjacent side lengths from F be x and y (in metres). The three distances from F to the other corners are x, y, and the diagonal d = √(x² + y²).
  2. We are given two distances from F: 3 and 5. These must be two of {x, y, √(x²+y²)}.
  3. Case check: 3 and 5 cannot both be diagonals; the diagonal is the largest of the three. If 3 and 5 were the two sides (x=3, y=5), then the remaining distance would be the diagonal √(3²+5²)=√34≈5.83 m.
  4. The other feasible assignment is that 3 is a side (say x=3) and 5 is the diagonal: √(3² + y²) = 5 ⇒ 9 + y² = 25 ⇒ y² = 16 ⇒ y = 4 m. The remaining distance is 4 m.
  5. Compare possibilities: 4 m is smaller than √34 ≈ 5.83 m, so the minimum possible distance from F to A is 4 m.

Final answer (with units): 4 m.

Teacher note to student: Great conceptual explanation! Next time write a quick labelled rectangle, show the key equation (√(3² + y²) = 5) and include the unit “m” beside your final number. That gives full credit and makes your reasoning visible to graders.

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