Find the equal side of an isosceles triangle — 13-year-old student (base 24, area 60)

Step-by-step solution and teacher feedback for a 13-year-old: using area to find height, splitting the isosceles triangle, then Pythagoras to get the equal side (13). Includes ACARA v9 mapping, rubric and Sailor Moon–style teacher comment.

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Teacher feedback and solution

Correctness: The answer is correct — each equal side has length 13.

Step-by-step solution

  1. Use the area formula: area = 1/2 × base × height. So 60 = 1/2 × 24 × h, giving h = 5.
  2. In an isosceles triangle the altitude from the apex bisects the base, so half the base = 24/2 = 12. The right triangle formed has legs 5 (height) and 12 (half-base).
  3. Apply Pythagoras in that right triangle: side^2 = 5^2 + 12^2 = 25 + 144 = 169, so side = √169 = 13.

Evaluation (mark out of 4)

  • Accuracy: 2/2 — correct final answer.
  • Method: 1/1 — appropriate method (use area to find height, then Pythagoras).
  • Communication: 1/1 — reasoning is clear and concise; could add a labelled diagram and units for completeness.

Total score: 4/4

Rubric (mapped to ACARA v9 — Measurement & Geometry)

  • Excellent (4): Correct answer, clear logical method using area and right-triangle reasoning, diagram labelled, units stated.
  • Proficient (3): Correct method and answer, minor presentation omissions (e.g. missing labels or units).
  • Developing (2): Partial method or small arithmetic errors; shows some understanding of area and right-triangle relationships.
  • Beginning (1): Incorrect method or misunderstanding of area/right-triangle concepts; limited reasoning.

ACARA v9 mapping: Measurement and Geometry — apply area formulas, represent and solve problems using right-triangle properties (Pythagoras), interpret and use measurements in problem-solving contexts.

Teacher comment (Sailor Moon cadence)

In the name of the Moon, brave learner, you have transformed numbers into starlight! Your steps show courage: you found the triangle’s height, halved the base to create a right triangle, and used the Pythagorean relation to reveal the hidden equal side. That method is sparklingly correct: 5^2 + 12^2 = 169, so the side is 13. Well done, soldier of geometry! For polish, write the height source: show how area = 1/2 × base × height gives 60 = 1/2 × 24 × h, so h = 5. Also label diagrams, include units, and explain why halving the base produces a right triangle (isosceles split down the altitude). For deeper training, try checking with the perimeter or exploring what happens if the area changes. To level up, describe why the altitude bisects the base in an isosceles triangle and practise similar problems with different bases and areas. Your working is efficient and mathematically sound; only small presentation details will make it perfect. Keep shining, sailor of maths — precision and neat reasoning will guide you to victory! Return with more solutions, and remember: clarity, units and a labelled diagram are your Moon Prism power for maths success. Always believe, Sailor!

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