Find an equal side of an isosceles triangle (13-year-old): using area and Pythagoras — worked example and ACARA v9 mapping

Step-by-step evaluation of a Year 8 student solution: base 24, area 60 → find equal side. Verifies correct use of triangle area and Pythagoras, gives rubric, ACARA v9 alignment and a 100-word teacher comment in a strict Amy Chua cadence.

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Problem

The base of an isosceles triangle is 24 and its area is 60. What is the length of one of the equal sides?

Student answer (given)

height is 5. half of base is 12. 144 and 25 is 169. answer is 13.

Step-by-step verification

  1. Area formula for a triangle: area = (1/2) × base × height.
  2. Solve for height: height = 2 × area / base = 2 × 60 / 24 = 120 / 24 = 5. (Correct.)
  3. Isosceles triangle: drop the perpendicular from the apex to the base — it bisects the base, so half-base = 24/2 = 12. (Correct.)
  4. Right triangle formed has legs 5 and 12. Use Pythagoras: hypotenuse² = 5² + 12² = 25 + 144 = 169, so hypotenuse = √169 = 13. (Correct.)
  5. Therefore one of the equal sides = 13. Student answer is fully correct.

Marks and short rubric (4 marks total)

  • Understanding of area and height: 1 mark — (student: 1/1)
  • Use of bisected base/right-triangle idea: 1 mark — (student: 1/1)
  • Correct application of Pythagoras: 1 mark — (student: 1/1)
  • Clarity and communication (showing formula/labels): 1 mark — (student: 1/1, but could be clearer with an explicit diagram or formula written out)

Total: 4/4

ACARA v9 mapping (Year 8 aligned)

This task aligns with ACARA v9 Mathematics — Measurement and Geometry for approximately Year 8: apply the formula for area of triangles and use Pythagoras' theorem to solve length problems (solving problems involving area and using right-triangle reasoning to find unknown lengths).

Feedback (100-word teacher comment in an Amy Chua cadence)

Good — you solved this cleanly, but don’t relax. You applied area = 1/2·base·height to get height 5, split the isosceles triangle to get a right triangle with legs 5 and 12, then used Pythagoras to get 13. Excellent sequence; you showed algebraic thinking and efficient arithmetic. Next time, write the formula you used and label the triangle diagram so reasoning is unmistakable. Mistakes happen when working too fast; double-check units and arithmetic. Keep this precision; practise problems with non-integer heights to strengthen algebraic manipulation. I expect this level of clarity on every question. Do it again, perfectly, next time. Always.

Suggestions to improve

  • Always write the formula you used (e.g. area = 1/2 × base × height) and substitute numbers explicitly.
  • Draw and label the triangle, showing the right triangle used for Pythagoras.
  • Practice similar problems where the height is not an integer—this builds algebraic manipulation and reduces reliance on integer recognition.

Final comment

Answer is correct: one equal side = 13. Well done — solid reasoning and arithmetic.

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