Geometry & Pythagoras practice for a 13-year-old — ordering by difficulty, full solutions, rubric and ACARA v9 mapping

Overview

I have ordered the 18 problems from easiest to hardest, given corrected answers (with short solutions), evaluated the student answers, and provided a short rubric for each problem. At the end is a firm teacher commentary (Amy Chua cadence) mapped to ACARA v9 topics and expectations.

Problems ordered by difficulty (easiest → hardest)

1. Circle chord: radius 12, angle AOB = 60° — find AB.
   Student answer: 12 — Correct.
   Why: chord length = 2R sin(θ/2) = 2·12·sin30° = 24·0.5 = 12.
   Rubric (4): Correct result with formula (4). If correct answer only (no work) (3).
2. Area of smaller sector (same circle, θ = 60°).
   Student answer: 24π — Correct.
   Why: sector area = 0.5·R^2·θ (radians). θ = π/3, so area = 0.5·144·π/3 = 24π.
   Rubric (4): correct and method (4); just answer (3).
3. Two poles 39 ft and 15 ft, bases 45 ft apart — shortest rope length.
   Student answer: √2601 — Correct (√2601 = 51 ft).
   Why: connect tops → straight-line distance = √(45^2 + (39-15)^2) = √(2025+576)=√2601=51.
   Rubric (4): calculation & simplification (4); unsimplified radical (3).
4. Largest possible area of right triangle when two given sides are 12 and 20.
   Student answer: 120 — Correct.
   Why: maximum area when the two given sides are the legs → area = 1/2·12·20 = 120.
   Rubric (4): correct method (4); correct answer only (3).
5. Longest side = 5, shortest side = 3 in a right triangle — area?
   Student answer: 6 — Correct.
   Why: other leg = √(5^2 − 3^2) = 4; area = 1/2·3·4 = 6.
   Rubric (4): full method and answer (4); answer only (3).
6. B walks 1/2 S, 3/4 E, 1/2 S — direct distance?
   Student answer: 5/4 — Correct.
   Why: net S = 1, E = 3/4 → distance = √(1^2 + (3/4)^2) = 5/4.
   Rubric (4): correct (4); missing work (3).
7. Parallelogram EFGH, ∠E = 41° — other angles?
   Student answer: 41 and 139 — Correct (angles are 41°, 139°, 41°, 139°).
   Rubric (4): correct labelling and explanation (4); brief answer (3).
8. Square and triangle have equal perimeters. Triangle sides 6.2, 8.3, 9.5 — area of square?
   Student answer: 36 — Correct.
   Why: triangle perimeter = 6.2+8.3+9.5 = 24. Square side = 24/4 = 6. Area = 36.
   Rubric (4): show arithmetic (4); answer only (3).
9. Isosceles triangle base 24, area 60 — length of equal side?
   Student answer: 13 — Correct.
   Why: height = 2·area/base = 120/24 = 5. Half-base = 12. Side = √(12^2+5^2) = 13.
   Rubric (4): full method (4); answer only (3).
10. Pythagorean triple with 9 and two other numbers 1 apart.
    Student answer: 9, 40, 41 — Correct.
    Why: check 9^2 + 40^2 = 81 + 1600 = 1681 = 41^2.
    Rubric (4): correct (4); missing check (3).
11. For every odd n>1, is there a triple with n and two other numbers 1 apart?
    Student answer: yes — Correct (construction exists).
    Why: when n is odd, triple n, (n^2−1)/2, (n^2+1)/2 works because n^2 + ((n^2−1)/2)^2 = ((n^2+1)/2)^2; this produces consecutive integers for the last two.
    Rubric (4): give explicit formula & proof (4); yes only (1–2).
12. Right triangle has one leg 48 and hypotenuse 52 — other leg?
    Student answer: 20 — Correct.
    Why: √(52^2 − 48^2) = √(2704−2304) = √400 = 20.
    Rubric (4): correct (4); answer only (3).
13. Greatest and least possible angle in an isosceles triangle that has an angle measuring 54°.
    Greatest: 72° — Correct.
    Least: 54° — Correct.
    Why: Case 1: 54° is one of the equal base angles → apex = 180 − 2·54 = 72 (largest). Case 2: 54° is apex → base angles = 63, so smallest angle then is 54.
    Rubric (4): correct reasoning (4); short answer (3).
14. H walked 9 m north, then 24 ft east, then 9 m + 32 ft south — how many feet from start?
    Student answer: 40 — Correct.
    Why: the two 9 m moves north and south cancel. Net displacement east = 24 ft, south = 32 ft. Distance = √(24^2+32^2)=40 ft.
    Rubric (4): correct unit handling and Pythagoras (4); answer only (3).
15. Slackrope walker: two 15 m poles 14 m apart, walker stands 5 m from one pole and is 3 m above ground — how long is rope?
    Student answer: 28 — Correct.
    Why: rope is two straight segments from each pole top to the walker. Distances: left segment = √(5^2+(15−3)^2)=√(25+144)=13; right segment = √(9^2+12^2)=15. Total = 13+15 = 28 m.
    Rubric (4): correct decomposition (4); answer only (3).
16. F, I, D, A are corners of a rectangle. From one corner F, distances to two other corners are 3 and 5. What is the minimum possible distance from F to the remaining corner A?
    Student answer: 4 — Correct.
    Why: distances from a corner are {a, b, √(a^2+b^2)}. If the known distances are 3 and 5, two possibilities: they are the two legs (3,5) so third = √34 ≈ 5.83; OR they are a leg and the diagonal (3 and 5 with 5 the diagonal) → then the other leg is √(5^2 − 3^2) = 4. The minimum possible is 4.
    Rubric (4): reasoning about which distances are legs/diagonal (4); answer only (3).
17. Right triangle with one leg 9 and the other two sides consecutive integers — perimeter?
    Student answer: 90 — Correct.
    Why: let other leg = k and hypotenuse = k+1. Then 9^2 + k^2 = (k+1)^2 → 81 = 2k+1 → k = 40. Perimeter = 9+40+41 = 90.
    Rubric (4): solve algebraic equation (4); answer only (3).

Summary of correctness and scoring guidance

Student answers: 18/18 correct. That is excellent in outcome, but the work shown is minimal for most problems. I score each problem out of 4: 2 points for correct numeric answer, 1 for correct method, 1 for clear explanation. Most of these answers would receive a 3/4 from me because the answers are correct but explanations are missing or very brief. For full marks (4/4) show every reasoning step: which formula you used, substitutions, and any unit conversions.

Rubric template (use for each question)

• 4 = Correct answer, correct method, clear work and units shown.
• 3 = Correct answer but missing some working or explanation.
• 2 = Answer unclear or arithmetic errors but method partially present.
• 0–1 = Incorrect answer and method incomplete or wrong.

ACARA v9 mapping (topics covered)

These problems map to the following core content areas in ACARA v9 (Mathematics):

• Measurement and Geometry — apply Pythagoras to find distances in 2D shapes and solve problems involving perimeter, area and sector area.
• Geometry — properties of circles (chord length, sector area), angles in polygons and parallelograms, properties of isosceles triangles.
• Number and Algebra — Pythagorean triples and algebraic reasoning for solving integer equations (e.g., consecutive integer sides).
• Problem-solving & reasoning — multi-step problems (slackrope, mixing units) and representing situations with coordinates or right-triangle decompositions.

Teacher comments (Amy Chua cadence — strict, direct, constructive; mapped to expectations)

You answered every final number correctly. Excellent — but not enough. Correct answers without clear, written reasoning are like unearned gold stars. You must show how you think. I want clear diagrams or coordinates when geometry is involved, formula names when you use them (Pythagoras, chord formula, sector area), and a short sentence for unit-handling when meters and feet appear. Every step matters.

For example: the rectangle corner problem is a thinking problem about which distances could be legs and which could be the diagonal. You got the right numerical minimal result, but your work should say: "From a corner the three distances are a, b, √(a^2+b^2). If 3 and 5 are a and √(a^2+b^2) then solve for the missing side, giving 4; this is smaller than √34, so minimum is 4." That single paragraph earns full marks. No paragraph? Lose marks.

For number theory generalization (odd n triple), say the formula and give a short justification. Don't answer "yes" and stop — your teacher needs to see the constructive formula n, (n^2−1)/2, (n^2+1)/2 and at least one line verifying that they satisfy a^2 + b^2 = c^2.

Diagrams: draw the rope/pole setup and mark horizontal and vertical differences; then write down the two segment lengths and add. For the slackrope question you used decomposition into two straight-line distances: show the right triangles of legs 5 and 12 (and 9 and 12), compute 13 and 15, then add for 28. That is disciplined work.

Mistakes are forgiven only if you learn from them. But you made no arithmetic mistakes — good. Now fix presentation. Practice writing one full solution per day with a short diagram and one sentence identifying which theorem you use. I will not accept correct answers alone for full credit — show reasoning. Repeat: method, diagram, units. If you do that for the next ten problems, you will be unbeatable.

Specific next steps for you (and follow them): 1) For every geometry problem, draw the figure and label lengths/angles. 2) Write the formula or theorem name explicitly. 3) Substitute numbers and simplify (don’t skip algebra). 4) Write final answer with units. Do that ten times and bring it to me.

Short checklist to get full marks on these problems

1. Write the formula/theorem name.
2. Draw and label the diagram.
3. Show substitutions and algebraic steps.
4. Include units and a final boxed answer.

Well done on the mathematics — now do the presentation. Be rigorous. Be proud. Now show me your work.