Why P’s 3-4-5 scaling fails for a 13-year-old: find the missing leg when one leg is 300 cm and hypotenuse 400 cm

Clear, step-by-step explanation for a Year 8 student showing why scaling a 3-4-5 triple the way P did is wrong, the correct use of Pythagoras, the exact and decimal length of the missing leg, and a 200-word ACARA v9-aligned teacher comment in Amy Chua cadence.

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Short answer: P’s method is wrong. The other leg has length √70000 = 100√7 ≈ 264.58 cm.

Step-by-step explanation (for a 13-year-old):

  1. P thought the triangle was a scaled 3-4-5 triangle. A 3-4-5 triangle has sides in the ratio 3:4:5, meaning if you multiply every side by the same scale factor k you get k·3, k·4, k·5.
  2. P noticed 300 = 3×100 and 400 = 4×100, but those two match the short leg and the other short leg of a 3-4-5 triangle, not a short leg and the hypotenuse. In a true 3-4-5 scaled triangle with k = 100 you would have sides 300, 400, and 500 where 500 is the hypotenuse. Here the hypotenuse is given as 400, so the sides are not in the 3:4:5 ratio.
  3. Use the Pythagorean theorem instead: if one leg a = 300 and hypotenuse c = 400, then the other leg b satisfies b^2 = c^2 − a^2 = 400^2 − 300^2 = 160000 − 90000 = 70000.
  4. So b = √70000. Simplify: √70000 = √(7×10000) = 100√7 ≈ 100×2.6458 = 264.58 cm (rounded to two decimal places).

Evaluation of the student's answer: The student wrote √70000 and correctly rejected 500 because the hypotenuse must be the longest side. That is essentially correct. To improve: simplify the radical to 100√7 and give a decimal approximation (≈264.58 cm). Also show the Pythagorean calculation step (400^2 − 300^2 = 70000) for full clarity.

ACARA v9 mapping (informal): Aligns with Year 8 Measurement and Geometry outcomes — apply Pythagoras' theorem to find the length of an unknown side of a right triangle.

200-word teacher comment (Amy Chua cadence):

Listen. You did the key thing: you noticed that 500 cannot be the missing side because the hypotenuse is the longest side. Good. But you must do more than notice — you must explain and calculate. Write the steps. Show 400^2 − 300^2 = 70000. Then simplify √70000 to 100√7 and give the decimal 264.58 cm. That is how evidence is shown. Do not rely on pattern-seeing alone. Spotting a 3-4-5 pattern is useful only when all three sides fit that ratio — here they do not. You must check that the hypotenuse corresponds to 5k, not 4k. If you want to be convincing, use the Pythagorean theorem every time. Neat, clear work. No shortcuts without justification. Practice three similar problems now: change the leg or hypotenuse and calculate. Do them until you can set up c^2 − a^2 quickly and simplify radicals correctly. You will not be forgiven for sloppy presentation. Precision matters.

Rubric (4-point scale):

  • Understanding (4): Student understands hypotenuse is longest and rejects 500 — score 4/4.
  • Method (4): Student used correct idea (wrote √70000) but didn’t show intermediate subtraction or simplification — score 3/4.
  • Accuracy (4): Numerical answer in radical form is correct — score 4/4.
  • Communication (4): Needs simplified radical and decimal approximation, and clearer steps — score 2.5/4.

Total: 13.5/16. Target next steps: always show Pythagoras subtraction step, simplify radicals, and supply a decimal approximation.

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