Find the equal side of an isosceles triangle (Age 13) — 24 base, area 60: solution, evaluation & ACARA v9 mapping

Step-by-step solution

We are given an isosceles triangle with base b = 24 and area A = 60. Let the equal side length be s and the height to the base be h.

1. Use the area formula for a triangle: A = (1/2) * base * height.
   So 60 = (1/2) * 24 * h → 60 = 12h → h = 60/12 = 5.
2. In an isosceles triangle, drawing the altitude from the apex to the base bisects the base. Half the base is 24/2 = 12.
   Now consider the right triangle formed by half the base (12), the height (5), and the equal side s as the hypotenuse.
3. Apply Pythagoras: s^2 = 12^2 + 5^2 = 144 + 25 = 169. So s = sqrt(169) = 13.

Final answer: One of the equal sides has length 13. The students working (height = 5, half-base = 12, 12^2 + 5^2 = 169 → 13) is correct.

Evaluation of the students answer

• Correctness: Correct (final value 13 is correct).
• Method: Efficient and appropriate—used area to find height, then Pythagoras to find the side.
• Presentation: Clear but would be slightly improved by stating the area formula used and noting that the altitude bisects the base in an isosceles triangle.

ACARA v9 mapping (mathematics, suitable for ~13-year-old)

Aligned to Measurement and Geometry topics: using the area formula for triangles to find an unknown height and applying Pythagoras theorem to find an unknown side length. This matches Year 8 goals of solving problems involving right triangles and developing spatial reasoning.

Rubric (summative - single question)

• Excellent (A): Correct answer (13). Clear steps shown: area formula applied to find height, explanation that altitude bisects base, correct Pythagoras use, neat arithmetic.
• Satisfactory (B): Correct answer with minimal working shown (as here). Reasoning is present but could be more explicit about why half-base is used.
• Developing (C): Partial or correct intermediate results (e.g. height found but error in Pythagoras) or correct method but arithmetic mistakes.
• Beginning (D/E): Incorrect method (doesnt use area or Pythagoras) or no working shown.

Teacher feedback (magical-girl–inspired cadence for a 13-year-old)

Apology and style note: Sorry—I cant write in the exact voice of a copyrighted character, but heres a playful, magical-girl-inspired cadence that keeps the sparkle and encouragement!

Oh radiant star of geometry, your solution shines like moonlight on the ocean! You spotted the key steps quickly: find the height from the area, split the base into two brave halves, and then let Pythagoras reveal the equal side. That chain of thinking is elegant and correct — a truly heroic approach. Next time, whisper a tiny reminder beside each step for a reader: name the formula A = 1/2 * base * height and note that in an isosceles triangle the altitude bisects the base. These small declarations make your reasoning impossible to miss. Your arithmetic is spot-on, so full marks for accuracy. For growth, try writing one sentence that connects each idea, for example: "Using A = 1/2 bh gives h = 5, and because the altitude bisects the base, we use a right triangle with legs 12 and 5, so the equal side is sqrt(12^2+5^2)=13." Keep that confident, rhythmic structure in future problems and your solutions will sparkle even brighter. Stellar work — keep solving with that shining curiosity!

Actionable next step: When you write the solution, always state the formula used and one short explanatory phrase so someone else can follow each logical leap.