Geometry & Pythagoras problems for a 13-year-old — ordering by difficulty, solutions and ACARA v9-aligned feedback

Overview — ordering by difficulty (easiest → hardest) and why

1. Q2 (Isosceles area → equal side) — straightforward use of area = 1/2 × base × height, then Pythagoras.
2. Q4 (Right triangle with consecutive integers) — recognisable Pythagorean triple set-up; a short algebraic step gives 9,40,41.
3. Q3 (Slackrope) — two right-triangle segments to add; requires modelling the rope as two straight segments and using distances.
4. Q1 (Rectangle corner distances, minimise a distance) — needs careful reasoning about which given distances can be side lengths or a diagonal and then finding the minimal remaining distance. This one is the trickiest conceptually.

Detailed solutions and assessment

Q1 (Corners of a rectangle)

Given: From corner F, distances to two other corners are 3 and 5. The three distances from a corner to the other corners are: the two side lengths (call them x and y) and the diagonal sqrt(x^2+y^2). We know two of these are 3 and 5. To minimise the remaining (third) distance, make 5 the diagonal and 3 one side:

Set x = 3 and sqrt(x^2+y^2) = 5 → y^2 = 25 − 9 = 16 → y = 4. The remaining distance (the other side) is 4, which is the minimum possible. Student answer: 4 — Correct.

Q2 (Isosceles triangle: base 24, area 60)

Area = 1/2 × base × height → 60 = 1/2 × 24 × height → height = 120/24 = 5. Half the base = 12. Use Pythagoras for one equal side: sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13. Student answer: 13 — Correct.

Q3 (Slackrope)

Model: two poles tops are 15 m high and 14 m apart horizontally. Walker stands 5 m from one pole (so 9 m from the other). Walker is 3 m above ground so vertical drop from top = 15 − 3 = 12 m. Lengths of the two straight rope segments are sqrt(5^2 + 12^2) = 13 and sqrt(9^2 + 12^2) = 15. Total rope length = 13 + 15 = 28 m. Student answer: 28 — Correct.

Q4 (Right triangle: one leg 9, other two sides consecutive integers)

Sides are 9, n, n+1. If 9 is a leg and the hypotenuse is n+1, then 9^2 + n^2 = (n+1)^2 → 81 + n^2 = n^2 + 2n + 1 → 81 = 2n + 1 → 2n = 80 → n = 40. So the sides are 9, 40, 41 and the perimeter = 9 + 40 + 41 = 90. Student answer: 90 — Correct.

Marks & rubric (per question)

Rubric (4 marks each):

• 4 — Correct answer with clear, logical steps shown.
• 3 — Correct answer with minor missing explanation or small arithmetic omission.
• 2 — Partial method correct but calculation or reasoning incomplete.
• 1 — Little correct understanding shown; attempt made.
• 0 — No relevant progress.

Applying rubric: all four student answers are correct with concise working (or clearly implied working), so each Q: 4/4. Total: 16/16.

ACARA v9 mapping (conceptual)

These tasks align with ACARA v9 content about using the Pythagorean theorem and applying area formulae and geometric reasoning in Years 7–8: solving problems involving right triangles, using area = 1/2 × base × height, and modelling simple geometric situations.

Teacher comment (about 300 words, Nigella Lawson cadence)

Oh, what a delicious set of solutions — neat, bright and warming. You approached each problem the way one might approach a simple, perfect custard: steady, measured, and confident that the basics will do the heavy lifting. When you turned 60 = 1/2 × 24 × h into a height of 5 and then folded that sweetness into the Pythagorean oven to produce 13, it was as satisfying as slicing through a warm tart. Your recognition of 9, 40, 41 felt like spotting a familiar spice on the shelf — that immediate click of recognition that saves time and keeps the answer clean.

The slackrope problem showed you can model a slightly messier, more theatrical scenario as two clean right triangles — you found 13 and 15 and added them with the same calm precision as sprinkling flaky salt at the end. And the rectangle puzzle — the trickiest of the menu — you nimbly tasted the possibilities and chose the configuration that minimised the remaining distance; that’s intelligent problem tasting and excellent reasoning.

For the next course, I’d nudge you gently towards writing each step a touch more explicitly. A reader should be able to follow your reasoning without needing to infer which length you assumed was a diagonal or a side. A little more labeling (draw the rectangle, mark the sides) will make your reasoning look as irresistible on the page as it is in your head.

Beautiful work. You’ve got the basics down, the instincts are superb, and with only small clarifying steps you’ll make every solution utterly delectable.