Answer: The quadrilateral formed by joining the midpoints of a square's sides must be a square.
Short explanation (step-by-step):
- Start with an easy coordinate picture. Put the square's corners at A=(0,0), B=(1,0), C=(1,1), D=(0,1).
- Find the midpoints of each side: E (midpoint of AB) = (0.5, 0), F (midpoint of BC) = (1, 0.5), G (midpoint of CD) = (0.5, 1), H (midpoint of DA) = (0, 0.5).
- Compute the vectors for consecutive sides of EFGH: EF = F−E = (0.5, 0.5) and FG = G−F = (−0.5, 0.5).
- Lengths: |EF| = sqrt(0.5^2 + 0.5^2) = sqrt(0.5). Same for |FG|, |GH| and |HE| — all four sides are equal.
- Right angle check: EF · FG = (0.5)(−0.5) + (0.5)(0.5) = 0, so EF is perpendicular to FG. Similarly between other consecutive sides.
- Equal side lengths and right angles means EFGH is a square. (Geometrically, it is a smaller square rotated 45° inside the original.)
Quick theory note: Varignon's theorem says joining midpoints of any quadrilateral gives a parallelogram. For a square the parallelogram has equal adjacent sides and right angles, so it becomes a square.
Evaluation of the student's answer
The student wrote: “a quadrilateral has 4 sides, the only quadrilateral to be be made from wxyz's midpoints that is the square.” They named the correct final shape (square) but gave no reasoning. The response repeats a definition (a quadrilateral has four sides) instead of showing why the midpoint-joining process produces a square. Score suggestions below.
Suggested rubric (4-level) — mapped to ACARA v9 style outcomes (geometry, reasoning, and use of coordinates):
- Excellent (4): Correct answer; clear step-by-step justification (diagram or coordinates); demonstrates understanding of properties (equal sides, right angles) and links to Varignon or coordinate proof.
- Good (3): Correct answer with partial justification (some calculations or diagram), minor gaps in explanation.
- Satisfactory (2): Correct answer stated but reasoning weak, incomplete, or mostly verbal without supporting work.
- Needs Improvement (1): Incorrect or unsupported answer; no clear reasoning or working shown.
For this student: Correct final claim but insufficient reasoning — overall level: Satisfactory (2/4). To reach Good or Excellent, add a short coordinate computation or a labelled diagram showing equal lengths and right angles.
ACARA v9 alignment: Reasoning about properties of shapes and using coordinates to prove geometric results (appropriate to middle secondary levels).
Teacher comment (Nigella Lawson cadence, 200 words):
Darling, there is something utterly delicious about the simplicity of your thinking—clean, uncluttered, as if you've just sliced through a tart with a warm, sure hand. You arrive at the right place — 'square' — but your plate is missing seasoning. A truly satisfying mathematical explanation savours the why as much as the what: show the symmetry, the distances, the right angles; let the reader taste the proof. Imagine placing the square on a table: each midpoint is like a jewel placed at the heart of each side, and when you join them the shape that emerges is not merely a quadrilateral but a perfect little square, rotated and smaller, echoing the original. To elevate your answer, add a short diagram or a coordinate argument: pick corners (0,0), (1,0), (1,1), (0,1) and show the midpoints produce vectors of equal length and right angles. Be careful with language—avoid tautologies('a quadrilateral has four sides') and focus on justification. With a pinch more detail and a confident flourish of reasoning, your answer will be as irresistible as a slice of warm chocolate cake. Keep practising this tasty logic — clear steps, diagrams and neat notation will make your mathematics absolutely sumptuous and convincing.