--- title: "Pythagoras' theorem Practice Questions & Worked Answers" description: "Find a missing side in a right-angled triangle, and test whether a triangle is right-angled at all. Free Years 8 to 9 practice with worked solutions for..." image: "https://nobleeducators.com/og-image.jpg" url: "https://nobleeducators.com/australia/practice/pythagoras-theorem" --- Your Australian school pathway · Noble Educators # Pythagoras' theorem practice questions and worked solutions Pythagoras' theorem relates the three sides of any right-angled triangle: the squares on the two shorter sides sum to the square on the hypotenuse. It is one of the few results students meet in middle years that they will still be using in Year 12, in trigonometry, in coordinate geometry and in vectors, so the time spent making it automatic pays back repeatedly. Book a free demo WhatsApp now ![Student working through a lesson in a notebook beside a laptop](https://nobleeducators.com/sat-redesign/support-student.webp) Live, one-to-one learning Noble Educators 1 ## Diagnose before drilling The first session finds what is actually breaking down — missing content, a weak method, or lost confidence — because practising the wrong thing wastes a term. 2 ## Plan to the real calendar Lessons aim at the student's next school task — a SAC, assessment, trial, NAPLAN window or exam block — not a generic syllabus order. 3 ## Practise under matching conditions Timed where the task is timed, written where it is written, with the calculator and formula rules the real assessment uses. 4 ## Report so parents can see it After lessons, parents get plain-language notes: what was covered, what improved, and the next piece of work — no jargon, no guesswork. ## Identifying the hypotenuse first In a² + b² = c², the letter c is always the hypotenuse: the side opposite the right angle, and always the longest of the three. Orientation makes no difference, and this is where diagrams catch students out, because textbook triangles sit neatly with the right angle at the bottom left while assessment triangles are rotated deliberately. The reliable habit is to find the right angle, then trace directly across the triangle to the side facing it, and label that side before writing anything else down. Students who label first make far fewer errors than students who assume the longest-looking side in the drawing is the hypotenuse, since diagrams are frequently not to scale. ## Adding versus subtracting When both shorter sides are known you add: c² = a² + b². When the hypotenuse and one shorter side are known you subtract: b² = c² − a². Adding in the second case is the single most common error in this topic, and it produces an answer longer than the hypotenuse, which is geometrically impossible. That impossibility is a free check. Before writing a final answer, ask whether the value you have found is shorter than the hypotenuse when it should be, or longer than both shorter sides when it is the hypotenuse. A student who applies this check catches almost every arithmetic slip without redoing the working. ## Pythagorean triples worth recognising Certain whole-number combinations appear constantly: 3-4-5, 5-12-13, 8-15-17 and 7-24-25, along with every multiple of them. A triangle with sides 9, 12 and 15 is simply 3-4-5 scaled by three, and spotting that lets you write the answer down rather than reaching for a calculator. Recognising triples is genuinely useful under time pressure, but it carries a trap. Students who have memorised them sometimes assume a triangle with sides 5, 12 and 14 must be right-angled because two of the numbers look familiar. Always verify with the theorem rather than by pattern-matching two of the three sides. ## Testing whether a triangle is right-angled The converse of the theorem is also true: if the squares on the two shorter sides sum to the square on the longest, the triangle is right-angled. This is a distinct skill from finding a missing side, and questions asking you to prove or disprove a right angle are testing the converse specifically. Do not confuse this with the triangle inequality, which says only that the two shorter sides must sum to more than the longest for a triangle to exist at all. That condition is about whether the shape can be drawn; Pythagoras is about whether it contains a right angle. Multiple-choice questions frequently offer the triangle inequality as a distractor. ## Where the theorem turns up later The distance formula in coordinate geometry is Pythagoras applied to the horizontal and vertical gaps between two points, and it is usually taught as a new formula rather than as the same idea in different clothing. Students who see the connection have one thing to remember instead of two. The theorem also underpins the exact values in trigonometry, the magnitude of a vector in senior courses, and three-dimensional problems where you apply it twice — once in the base plane and again using that result as one side of a vertical triangle. Two-step three-dimensional questions are a standard senior assessment item. ## Practise it now 5 questions Choose an answer and the method appears straight away. Nothing is recorded and there is no time limit — work through them at whatever pace is useful, and read the working even on the ones you get right. 1. Question 1. A right-angled triangle has shorter sides of 6 cm and 8 cm. How long is the hypotenuse?10 cm14 cm48 cm√14 cm 2. Question 2. A right-angled triangle has a hypotenuse of 13 m and one shorter side of 5 m. Find the other side.12 m8 m18 m√194 m 3. Question 3. Is a triangle with sides 9 cm, 12 cm and 15 cm right-angled?Yes, because 9² + 12² = 15²No, because the sides are all differentYes, because 9 + 12 > 15There is not enough information 4. Question 4. A square has sides of 5 cm. How long is its diagonal, to two decimal places?7.07 cm10.00 cm5.00 cm25.00 cm 5. Question 5. In a² + b² = c², which side does c always represent?The hypotenuse, opposite the right angleThe longest of the two shorter sidesThe vertical sideAny side you choose Years 8 to 9 · Australian Curriculum v9.0 content descriptors AC9M9M03, AC9M9SP01. Descriptor codes are published by ACARA and are quoted so you can check this practice against your school’s scope and sequence. Noble Educators is an independent tutoring provider and is not affiliated with ACARA. Common questions ## What families ask about this ### What year do Australian students learn Pythagoras' theorem? It is introduced in Year 8 under the Australian Curriculum and developed through Year 9, where students apply it to problems in context and combine it with trigonometry. Some schools introduce it late in Year 7 in extension classes. ### Does the theorem work for triangles without a right angle? No, it applies only to right-angled triangles. For other triangles you need the cosine rule, which is effectively Pythagoras with a correction term for the angle and reduces to it exactly when that angle is ninety degrees. ### Should answers be left as surds or rounded? Follow the wording of the question. Exact form means leaving a surd such as 5√2 unevaluated, while a request for two decimal places means rounding at the end only. Rounding partway through and then continuing is a common source of lost accuracy marks. ### Is Pythagoras' theorem tested in NAPLAN? It can appear in Year 9 numeracy, since the theorem is a Year 8 topic and NAPLAN assesses content students have already covered. Year 7 numeracy does not assume it, so a Year 7 student meeting it in practice material is working ahead. ## What sessions cost Every year level has a published per-session rate in AUD, with no lock-in contract. Year 1$12 AUD per session Year 7$18 AUD per session Year 12$26 AUD per session Booking a block reduces the per-session rate: 5% from 8 sessions, 8% from 12 sessions, 12% from 15 sessions, 15% from 20 sessions. At 12 sessions, Year 1 works out to $11.04 per session. The first lesson is free and there is no lock-in contract. 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[See the full cost comparison](https://nobleeducators.com/australia#compare) For parents ## Clarity before you book, visibility after every lesson. ## What to share before the first lesson The tutor match works from evidence, not a job title. Anything from the list below gives the first session a concrete starting point instead of a cold diagnostic. - The most recent school report or assessment result - The current task sheet, assessment notice or study design topic - Any teacher feedback, and the topic where confidence broke Send it through the enquiry form ## What you will see after every lesson Progress a parent cannot see is progress they cannot support. 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