Right triangle legs problem for 10-year-old — explain why scaling a 3-4-5 triple fails here and find the correct third side (Student age 10)

Clear, step-by-step explanation for a 10-year-old showing why P's scaling of a 3-4-5 triple is wrong when one leg is 300 cm and hypotenuse 400 cm. Includes exact and decimal answers, evaluation of the student's response, a rubric, and firm teacher feedback.

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Quick answer: P's method doesn't work because the hypotenuse must be the longest side. If you scaled the 3-4-5 triple by 100 you would get sides 300, 400, 500, but that makes 500 the hypotenuse, not 400. The correct third side is sqrt(400^2 - 300^2) = sqrt(160000 - 90000) = sqrt(70000) = 100·sqrt(7) ≈ 264.58 cm.

Step-by-step for a 10-year-old:

  1. Remember: in a right triangle the hypotenuse is the longest side.
  2. P thought: 300 = 3×100 and 400 = 4×100, so maybe the triangle is 3-4-5 scaled by 100 giving third side 5×100 = 500. But if one side were 500, that would be bigger than 400, so 400 could not be the hypotenuse. That is a contradiction.
  3. The Pythagorean theorem says (hypotenuse)^2 = (leg1)^2 + (leg2)^2. Rearranged: (other leg)^2 = (hypotenuse)^2 - (one leg)^2.
  4. Compute: other leg = sqrt(400^2 - 300^2) = sqrt(160000 - 90000) = sqrt(70000) = 100·sqrt(7) ≈ 264.58 cm.

Evaluate the student's answer: The student wrote "sqrt 70000" and correctly noted that the hypotenuse must be the longest side so there cannot be a 500 cm side. That is essentially correct. To be clearer and complete, the student should write the simplified exact form 100·sqrt(7) and give a decimal approximation ≈ 264.6 cm, and show the subtraction step 400^2 - 300^2 = 70000.

Rubric (4 levels):

  • 4 (Excellent): Correct exact form (100·sqrt(7)), correct decimal, clear Pythagoras steps, and explains why 3-4-5 scaling fails.
  • 3 (Good): Gives correct result (exact or decimal) and brief correct reasoning, missing small steps.
  • 2 (Developing): Correct idea but arithmetic or form incomplete (e.g. gives sqrt(70000) without simplification or explanation).
  • 1 (Needs Improvement): Incorrect result or misunderstanding of hypotenuse/leg relationship.

Teacher feedback (firm, direct tone; emulating a strict tutor style, not the public figure):

I like that you noticed the hypotenuse must be longest — very good. But show your work: write 400^2 - 300^2 = 70000 so everyone sees the steps. Simplify sqrt(70000) to 100·sqrt(7) and include a decimal so we know the size: about 264.6 cm. Don’t stop at a statement; prove it on paper. Next time, check the largest side first before trying to match a triple. You are on the right track; be precise and complete. Practice three similar problems where one leg and the hypotenuse are given; use Pythagoras every time until it feels natural.

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