Isosceles triangle problem (Age 13): find equal side when base = 24 and area = 60 — step-by-step solution

Step-by-step solution for a 13-year-old: using area to find height, halving the base, applying Pythagoras to get the equal side = 13. Includes evaluation, rubric and a 100-word ACARA v9-mapped teacher comment in Nigella Lawson cadence.

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Step-by-step solution

  1. Area formula for a triangle: Area = 1/2 × base × height.
    So 60 = 1/2 × 24 × height.
  2. Solve for height: 60 = 12 × height, so height = 60 ÷ 12 = 5.
  3. In an isosceles triangle the altitude from the apex bisects the base, so half the base = 24 ÷ 2 = 12.
  4. Now use Pythagoras on the right triangle formed: equal side^2 = 12^2 + 5^2 = 144 + 25 = 169.
    Therefore the equal side = sqrt(169) = 13.

Final answer: one of the equal sides has length 13 (units).

Evaluation of the student answer

The student wrote: "height is 5. half of base is 12. 144 and 25 is 169. answer is 13." This is correct. The method is exactly right: they used the area to find the height, halved the base, and applied Pythagoras to get 13. Minor improvement: state units (e.g. cm) if given, and write the steps slightly more clearly with labels (height = 5). Awarded marks: 4/4.

Simple marking rubric (4 marks total)

  • Method: Correct general approach (use area to find height and Pythagoras) — 1 mark (awarded 1).
  • Calculation: Correct arithmetic to find height and half-base — 1 mark (awarded 1).
  • Application of Pythagoras: correct set-up and computation — 1 mark (awarded 1).
  • Communication: clear steps and final answer (including units if required) — 1 mark (awarded 1, with suggestion to add units next time).

ACARA v9 mapping: This task maps to Year 8 geometry skills — using area formulas and Pythagoras to solve problems involving triangles (applying measurement and reasoning).

My dear, what a delicious little solution you've plated here. You measured the triangle's appetite for area, fed it with base and area to find a height of 5 — perfectly cooked. Then you sliced the base in half, warmed the numbers in the pan of Pythagoras, and emerged with the crisp, golden 13. It's elegant and simple, like a perfect lemon tart: balanced, precise and satisfying. Continue to write each step as you did, and remember to label the height and the right triangle formed. Little garnishes: include units and a final sentence stating the answer. Bravo — beautifully done, truly.

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