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The other two sides are $XY=8\\text{ m}$ and $YZ=15\\text{ m}$.","highlights":[{"text":"The right angle is at vertex $Y$.","location":"specification"},{"text":"The side $XY$ is $8\\text{ m}$.","location":"specification"},{"text":"The side $YZ$ is $15\\text{ m}$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=formula}{XY^2+YZ^2}=XZ^2$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=value_sum}{8^2+15^2}}=XZ^2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"value_sum"}]},"order":1,"title":"Apply Pythagoras","description":"For a right-angled triangle, the squares of the two shorter sides add to the square of the hypotenuse. Substitute both given lengths; writing this correctly earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$XZ^2=\\color{@orange}{\\htmlData{anchor=squares}{8^2+15^2}}$"},{"latex":"$$XZ^2=\\color{@orange}{\\htmlData{anchor=values}{64+225}}$"},{"latex":"$$XZ^2=\\color{@green}{\\htmlData{anchor=total}{289}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"values"},{"color":"green","style":"text-color","anchor":"total"}],"connections":[{"to":"values","from":"squares","color":"orange","label":"evaluate squares"}]},"order":2,"title":"Evaluate the squares","description":"Square each shorter side, then add. Reaching the correct total earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$XZ^2=\\color{@green}{\\htmlData{anchor=total}{289}}$"},{"latex":"$$XZ=\\sqrt{\\color{@green}{\\htmlData{anchor=root_value}{289}}}$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=answer}{XZ=17\\text{ m}}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"root_value"},{"color":"sky","style":"text-color","anchor":"answer"}],"connections":[{"to":"root_value","from":"total","color":"green","label":"take square root"}]},"order":3,"title":"Take the square root","description":"Use the positive square root because a length cannot be negative. Include the unit in the final answer; the correct length earns the accuracy mark."}]}],"generatedAt":"2026-07-19T16:30:26.214Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"110a2095-b742-4587-883a-2b782441873c:pythagoras_theorem","label":"Pythagoras' Theorem","steps":[{"type":"text","order":0,"title":"Identify the given values","content":"We are given a right-angled triangle $XYZ$ where the right angle is at $Y$.\n\nThe lengths of the two legs are:\n- $XY = 8\\text{ m}$\n- $YZ = 15\\text{ m}$\n\nWe need to find the length of the hypotenuse, $XZ$, which is the side opposite the right angle.","highlights":[{"text":"The side $XY$ is $8\\text{ m}$.","location":"specification"},{"text":"The side $YZ$ is $15\\text{ m}$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:"},{"latex":"$$$\\htmlData{anchor=hyp}{\\color{@sky}{XZ}}^2 = \\htmlData{anchor=side1}{\\color{@amber}{XY}}^2 + \\htmlData{anchor=side2}{\\color{@purple}{YZ}}^2$$"},{"text":"Substituting our given lengths into the formula:"},{"latex":"$$$XZ^2 = \\htmlData{anchor=sub1}{\\color{@amber}{8}}^2 + \\htmlData{anchor=sub2}{\\color{@purple}{15}}^2$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"hyp"},{"color":"amber","style":"text-color","anchor":"side1"},{"color":"purple","style":"text-color","anchor":"side2"}],"connections":[{"to":"sub1","from":"side1","color":"amber","label":null,"style":"solid"},{"to":"sub2","from":"side2","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Set up Pythagoras' Theorem","description":"Substitute the lengths of the two known sides into Pythagoras' theorem."},{"type":"text","order":2,"title":"Evaluate the squares and sum","content":"Now we calculate the squares of the two sides and add them together:\n\n$$\\begin{aligned} XZ^2 &= 8^2 + 15^2 \\\\ XZ^2 &= 64 + 225 \\\\ XZ^2 &= 289 \\end{aligned}$$"},{"type":"text","order":3,"title":"Calculate the hypotenuse length","content":"To find $XZ$, take the square root of $289$:\n\n$$XZ = \\sqrt{289} = 17\\text{ m}$$\n\nThis gives us our final answer of **$17\\text{ m}$**.","calculator":[{"gdc":{"expression":"\\sqrt{289}","instructions":"Use the square root key, enter 289, and press EXE."},"ti84":{"expression":"\\sqrt{289}","instructions":"Press [2nd] [x²] (which is the √ symbol), type 289, and press [ENTER]."},"desmos":{"expression":"\\sqrt{289}","instructions":"Type 'sqrt(289)' into the expression bar."},"description":"You can find the square root of 289 directly on your calculator."}]}]},{"id":"110a2095-b742-4587-883a-2b782441873c:pythagorean_triple","label":"Pythagorean Triple Recognition","steps":[{"type":"text","order":0,"title":"Identify the given sides","content":"We are given a right-angled triangle $XYZ$ with the right angle at vertex $Y$. This means $XZ$ is the hypotenuse, and the two legs are:\n\n* $XY = 8\\text{ m}$\n* $YZ = 15\\text{ m}$\n\nTo find the hypotenuse $XZ$, we can look for a standard integer relationship between these lengths.","highlights":[{"text":"The side XY is 8 m","location":"specification"},{"text":"The side YZ is 15 m","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Given legs: } \\htmlData{anchor=side1}{\\color{@sky}{8}} \\text{ and } \\htmlData{anchor=side2}{\\color{@amber}{15}}$"},{"latex":"$$\\text{Pythagorean Triple: } (\\htmlData{anchor=t1}{\\color{@sky}{8}}, \\; \\htmlData{anchor=t2}{\\color{@amber}{15}}, \\; \\htmlData{anchor=t3}{\\color{@purple}{17}})$"},{"latex":"$$\\text{Hypotenuse } XZ = \\htmlData{anchor=side3}{\\color{@purple}{17}}\\text{ m}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"side1"},{"color":"amber","style":"text-color","anchor":"side2"},{"color":"purple","style":"text-color","anchor":"t3"}],"connections":[{"to":"t1","from":"side1","color":"sky","label":null,"style":"solid"},{"to":"t2","from":"side2","color":"amber","label":null,"style":"solid"},{"to":"side3","from":"t3","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Recognize the triple","description":"A Pythagorean triple consists of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. By recognizing the legs, we can immediately identify the hypotenuse."},{"type":"text","order":2,"title":"Verify with Pythagoras","content":"To ensure we secure all the marks on the exam, let's write down the verification using Pythagoras' theorem:\n\n$$XZ^2 = XY^2 + YZ^2$$\n\nSubstitute the values to show the working:\n\n$$\\begin{aligned} XZ^2 &= 8^2 + 15^2 & & \\color{gray}{\\text{[M1 for applying Pythagoras]}} \\\\\\\\ XZ^2 &= 64 + 225 \\\\\\\\ XZ^2 &= 289 & & \\color{gray}{\\text{[M1 for evaluating squares]}} \\\\\\\\ XZ &= \\sqrt{289} = 17\\text{ m} & & \\color{gray}{\\text{[A1 for final answer]}} \\end{aligned}$$\n\nOur triple recognition is perfectly verified!"}]}],"generatedAt":"2026-06-12T16:08:14.141Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"39062c59-6289-4106-9977-bfef86254c78","revisionId":"74804e0e-fcec-43ac-8db5-eccb85a9e08c","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"0253c0d2-4002-4b20-9172-578474aec172","label":"a","content":"A right-angled triangle $DEF$ is shown. The right angle is at vertex $E$. The horizontal side $DE$ is $7\\text{ cm}$. The vertical side $EF$ is $24\\text{ cm}$.\n\nCalculate the length of $DF$.","markscheme":"- Applies Pythagoras' theorem: $7^2 + 24^2 = DF^2$ M1\n- Evaluates squares and sums: $49 + 576 = 625$ M1\n- Final answer 25 A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Applies Pythagoras' theorem correctly: $7^2 + 24^2 = DF^2$","code":"M"},{"id":"p2","marks":1,"description":"Evaluates squares and sums: $49 + 576 = 625$","code":"M"},{"id":"p3","marks":1,"description":"Calculates the final length as 25","code":"A"}],"finalAnswers":["25"],"generatedAt":"2026-05-06T14:03:15.961Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"pythagoras","label":"Use Pythagoras' theorem","steps":[{"type":"text","order":0,"title":"Identify the hypotenuse","content":"The hypotenuse is the side opposite the right angle and is always the longest side. Since the right angle is at $E$, the hypotenuse is $DF$. The other two sides are $DE=7\\text{ cm}$ and $EF=24\\text{ cm}$.","highlights":[{"text":"The right angle is at vertex $E$.","location":"specification"},{"text":"The horizontal side $DE$ is $7\\text{ cm}$.","location":"specification"},{"text":"The vertical side $EF$ is $24\\text{ cm}$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=formula}{\\color{@sky}{a^2+b^2=c^2}}$"},{"latex":"$$\\htmlData{anchor=substitution}{\\color{@sky}{7^2+24^2=DF^2}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"formula"},{"color":"sky","style":"text-color","anchor":"substitution"}],"connections":[{"to":"substitution","from":"formula","color":"sky","label":"substitute side lengths"}]},"order":1,"title":"Apply Pythagoras","description":"The hypotenuse is the side opposite the right angle, so it is DF. Substitute the two perpendicular side lengths into Pythagoras' theorem. This earns the method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=start_eq}{7^2+24^2=DF^2}$"},{"latex":"$$\\htmlData{anchor=squares_eq}{49+576=DF^2}$"},{"latex":"$$\\htmlData{anchor=sum_eq}{625=DF^2}$"}]},"order":2,"title":"Evaluate the squares","description":"Evaluate 7² + 24² as 49 + 576 = 625. Reaching this correct total earns the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=sq_eq}{DF^2=625}$"},{"latex":"$$\\htmlData{anchor=root_eq}{DF=\\sqrt{625}}$"},{"latex":"$$\\htmlData{anchor=answer}{DF=25\\text{ cm}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"root_eq"},{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":3,"title":"Take the square root","description":"Take the positive square root because a length cannot be negative. This gives the accuracy mark (A1)."}]}],"generatedAt":"2026-07-19T18:42:17.743Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0253c0d2-4002-4b20-9172-578474aec172:pythagoras","label":"Pythagoras' Theorem","steps":[{"type":"text","order":0,"title":"Identify the given values","content":"We are given a right-angled triangle $DEF$ with:\n- The right angle at vertex $E$\n- Horizontal side $DE = 7\\text{ cm}$\n- Vertical side $EF = 24\\text{ cm}$\n\nSince the right angle is at $E$, the side opposite to it, $DF$, is the hypotenuse. We want to find the length of $DF$.","highlights":[{"text":"horizontal side DE is 7 cm","location":"specification"},{"text":"vertical side EF is 24 cm","location":"specification"},{"text":"Calculate the length of DF","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$a^2 + b^2 = c^2$$"},{"latex":"$$$\\htmlData{anchor=side-a}{\\color{@sky}{7}}^2 + \\htmlData{anchor=side-b}{\\color{@amber}{24}}^2 = \\htmlData{anchor=hyp}{\\color{@purple}{DF}}^2$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"side-a"},{"color":"amber","style":"text-color","anchor":"side-b"},{"color":"purple","style":"text-color","anchor":"hyp"}],"connections":[{"to":"hyp","from":"side-a","color":"sky","label":null,"style":"dashed"},{"to":"hyp","from":"side-b","color":"amber","label":null,"style":"dashed"}]},"order":1,"title":"Apply Pythagoras' theorem","description":"Pythagoras' theorem states that in any right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. We substitute our known values into the formula to earn our first mark."},{"type":"text","order":2,"title":"Evaluate squares and sum","content":"Now, we square each number and add them together:\n\n$$\\begin{aligned} 7^2 &= 49 \\\\ 24^2 &= 576 \\end{aligned}$$\n\nSumming these values together:\n\n$$DF^2 = 49 + 576$$\n$$DF^2 = 625$$\n\nThis calculation earns the second mark.","calculator":[{"gdc":{"expression":"7^2 + 24^2","instructions":"Type 7^2 + 24^2 and press [EXE] or [ENTER]."},"ti84":{"expression":"7² + 24²","instructions":"Type 7, press [x²], press [+], type 24, press [x²], and then press [ENTER]."},"desmos":{"expression":"7^2 + 24^2","instructions":"Type 7^2 + 24^2 in the input bar."},"description":"Use your calculator to easily square and sum the sides in one step."}]},{"type":"text","order":3,"title":"Find the square root","content":"To find the length of $DF$, take the square root of both sides:\n\n$$DF = \\sqrt{625}$$\n$$DF = 25\\text{ cm}$$\n\nThis final step gives us our third and final mark.","calculator":[{"gdc":{"expression":"√(625)","instructions":"Type sqrt(625) or use the square root template, then press [ENTER]."},"ti84":{"expression":"√(625)","instructions":"Press [2nd] then [x²] (to get √), type 625, and press [ENTER]."},"desmos":{"expression":"\\sqrt{625}","instructions":"Type sqrt(625) to get the result."},"description":"Calculate the square root of 625 to find the final hypotenuse length."}]}]},{"id":"0253c0d2-4002-4b20-9172-578474aec172:pythagorean_triple","label":"Pythagorean Triple Recognition","steps":[{"type":"text","order":0,"title":"Identify the given sides","content":"We are given a right-angled triangle $DEF$ with the right angle at $E$.\n\n* The horizontal side (leg) is $DE = 7\\text{ cm}$\n* The vertical side (leg) is $EF = 24\\text{ cm}$\n* We need to find the length of the hypotenuse, $DF$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Pythagorean Triple: } (\\htmlData{anchor=leg1}{\\color{@sky}{7}}, \\; \\htmlData{anchor=leg2}{\\color{@amber}{24}}, \\; \\htmlData{anchor=hyp}{\\color{@purple}{25}})$"},{"latex":"$$\\text{Pythagoras' Equation: } \\htmlData{anchor=a-eq}{\\color{@sky}{a}}^2 + \\htmlData{anchor=b-eq}{\\color{@amber}{b}}^2 = \\htmlData{anchor=c-eq}{\\color{@purple}{c}}^2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"leg1"},{"color":"amber","style":"text-color","anchor":"leg2"},{"color":"purple","style":"text-color","anchor":"hyp"}],"connections":[{"to":"a-eq","from":"leg1","color":"sky","label":"leg","style":"solid"},{"to":"b-eq","from":"leg2","color":"amber","label":"leg","style":"solid"},{"to":"c-eq","from":"hyp","color":"purple","label":"hypotenuse","style":"solid"}]},"order":1,"title":"Recognize the Pythagorean triple","description":"Recall that a **Pythagorean triple** consists of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. Knowing the well-known primitive triple $(7, 24, 25)$ lets us instantly identify the hypotenuse."},{"type":"text","order":2,"title":"Verify using Pythagoras","content":"To ensure we secure all the working marks on an exam, let's write out the calculation to verify our recognized triple:\n\n$$\\begin{aligned} DE^2 + EF^2 &= DF^2 \\\\ 7^2 + 24^2 &= DF^2 \\\\ 49 + 576 &= DF^2 \\\\ 625 &= DF^2 \\end{aligned}$$\n\nNow, taking the square root:\n\n$$DF = \\sqrt{625} = 25\\text{ cm}$$"},{"type":"text","order":3,"title":"State the final answer","content":"The length of $DF$ is $25\\text{ cm}$."}]}],"generatedAt":"2026-06-12T14:14:44.972Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":86},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/review/d1e62387-19b1-43c3-8104-1602b40e9918.mp4","durationSeconds":98,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/review/d1e62387-19b1-43c3-8104-1602b40e9918.jpg"}}],"subjectCode":"gcse_maths__wjec","summaries":[{"id":"0c776fa1-b652-4fa3-a413-27e90fd7de92","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0eb90c30-bf48-4480-aecb-d9aa3ba97db6","difficultyBand":"hard","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1bc2117d-6c77-4f77-ae0f-2d347b267bba","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6ea502a7-c3b0-403c-bae3-736213ee9d2d","difficultyBand":"hard","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6edfe765-f5c8-4f6d-bf89-f42899f49348","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"70a5f944-fcb0-4829-a1b4-7f4e899f595d","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"72055b6b-1c5b-4f97-bd7a-83f3d971038c","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"7508066f-5e6d-45cf-9874-f11ab5569ec9","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"82ed226d-6022-4947-8c2a-8c0ea3247836","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"8a2ad504-9c72-4c25-a3d0-af736455ffd9","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"909cfeb4-f19d-4386-99e6-22dec19f3b11","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9269555b-ea7f-4ccf-b172-082d2496e668","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a3f8775f-b8d5-4295-97a3-9a7b6760aebc","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a77acdd7-f54a-4745-b9ee-c26cf52cc9f2","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"aec56d01-5018-4dc5-a9dd-75c4c665a2d0","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b01074f3-244d-4883-9a87-09ce13adbba6","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d37e0969-abe5-48c9-868f-3a3da170da44","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"edd2b957-09e1-44bb-b2e4-c38d6dce83ca","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b0650435-f11e-45c3-b786-b9d1c9f6db4b","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0e039239-fd96-4549-b61f-6f5bf76ab060","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"11a07cb2-b3a0-4458-9cab-4b7d66888bec","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1bf5bd89-d40f-4c33-930a-6e7ca0674ac0","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"33a642a7-c988-4d7d-aacd-3a4851e290ab","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3a3161f8-38af-4a18-bc3c-c713d735e3be","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3edaaa3c-6185-4513-9f71-753dcfafb7a1","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"52f31c97-2ce9-44ca-b293-89e0739301bc","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"54489571-2af1-4bf6-9729-b5aac9dfc696","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"57617327-3a4f-4fbf-9a99-eb685da45a21","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5ad35bf8-229c-419d-83dc-99ba48f5ac85","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"85067d44-6f12-4347-99ec-51f7c614088a","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"8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