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[
{
"slide": 1,
"fragments": [
{
"fragment_index": -1,
"text_description": "Definition of Circle",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Circle",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "A circle is the set of all points that are exactly the same distance (the radius) from one fixed point (the centre).",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Find the centre first; any straight line from it to the curve is a radius.",
"image_description": ""
}
]
},
{
"slide": 2,
"fragments": [
{
"fragment_index": -1,
"text_description": "Arc vs Chord",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Circle diagram showing an arc (curved) and a chord (straight).",
"image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/presentation_images/asset.sparkl.me/pb/presentation/2558/images/c28393a344f1aca89c22af0301c0dfb3.png"
},
{
"fragment_index": 2,
"text_description": "What’s the difference?\nBoth terms describe how the same two points on a circle are connected.\nKnowing which is which lets you talk about circle parts accurately.\nKey Points:\nArc – curved slice of the circumference between two points.\nChord – straight line segment joining the same two points inside the circle.\nEvery diameter is a chord; it is simply the longest possible one.",
"image_description": ""
}
]
},
{
"slide": 3,
"fragments": [
{
"fragment_index": -1,
"text_description": "Arc Length Formula",
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},
{
"fragment_index": 1,
"text_description": "Central angle θ and radius r define the arc.",
"image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/img/lp/study_content/lp/1/11/4/182/651/1249/1237/21-5-09_LP_Vandana_Phy_1.11.4.4.2.4_srav_SS_html_m162ce79c.png"
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"text_description": "Formula (θ in degrees)\nArc length is the distance along the curved edge between two points on a circle.\nUse \\( \\text{Arc Length} = 2\\pi r \\times \\frac{\\theta}{360^\\circ} \\) to calculate it when θ is measured in degrees.\nKey Points:\nθ is the central angle in degrees.\n\\( \\frac{\\theta}{360^\\circ} \\) gives the fraction of the full circle.\nMultiply this fraction by the circumference \\( 2\\pi r \\).",
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}
]
},
{
"slide": 4,
"fragments": [
{
"fragment_index": -1,
"text_description": "Multiple Choice Question\nSubmit Answer\nconst correctOption = 1;\n const answerCards = document.querySelectorAll('.answer-card');\n const submitBtn = document.getElementById('submitBtn');\n const feedbackCorrect = document.getElementById('feedbackCorrect');\n const feedbackIncorrect = document.getElementById('feedbackIncorrect');\n\n let selectedOption = null;\n\n answerCards.forEach((card, index) => {\n card.addEventListener('click', () => {\n answerCards.forEach(c => c.classList.remove('border-blue-500', 'bg-blue-50'));\n card.classList.add('border-blue-500', 'bg-blue-50');\n selectedOption = index;\n });\n });\n\n submitBtn.addEventListener('click', () => {\n if (selectedOption === null) return;\n\n if (selectedOption === correctOption) {\n feedbackCorrect.classList.remove('hidden');\n feedbackIncorrect.classList.add('hidden');\n } else {\n feedbackIncorrect.classList.remove('hidden');\n feedbackCorrect.classList.add('hidden');\n }\n });",
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{
"fragment_index": 1,
"text_description": "Question\nSelf-check: Which statement is true about circles?",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "1\n(a) Every arc is a chord.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "2\n(b) A chord can pass through the centre.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "3\n(c) Arc length is always 2πr.",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "4\n(d) A radius is longer than a diameter.",
"image_description": ""
},
{
"fragment_index": 6,
"text_description": "Hint:\nThink of the chord that becomes the diameter when it passes through the centre.",
"image_description": ""
},
{
"fragment_index": 7,
"text_description": "Correct!\nA chord through the centre is called the diameter, the longest chord of the circle.",
"image_description": ""
},
{
"fragment_index": 8,
"text_description": "Incorrect\nReview circle facts: chords join two points on the circle, and the diameter is the special chord through the centre.",
"image_description": ""
}
]
},
{
"slide": 5,
"fragments": [
{
"fragment_index": -1,
"text_description": "Sectors Explained",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Circle showing a minor and major sector",
"image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/img/lp/study_content/editlive_lp/75/2012_06_18_11_10_41/5.png"
},
{
"fragment_index": 2,
"text_description": "Major vs Minor Sector\nA sector is a slice of a circle bordered by two radii and their arc.\nKnowing the size of the slice helps you spot which type it is.\nKey Points:\nSmaller slice (central angle < 180°) → minor sector.\nRemaining part of the circle → major sector.",
"image_description": ""
}
]
},
{
"slide": 6,
"fragments": [
{
"fragment_index": -1,
"text_description": "Segments Explained",
"image_description": ""
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{
"fragment_index": 1,
"text_description": "Chord AB creates a minor segment.",
"image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/img/lp/study_content/lp/1/12/15/240/942/1882/2090/LP_1.12.1.11.2.5_UV_SU_SS_html_13fd7bfd.gif"
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{
"fragment_index": 2,
"text_description": "What is a Segment?\nA segment is the part of a circle cut off by a chord.\nIt looks like a slice or bite removed, bounded by the chord and the arc.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Key Points:\nChord – straight line joining two points on the circle.\nCurved edge of the segment is an arc.\nSegments are named minor (small) or major (large).",
"image_description": ""
}
]
},
{
"slide": 7,
"fragments": [
{
"fragment_index": -1,
"text_description": "Area of Segment",
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{
"fragment_index": 1,
"text_description": "Circle segment illustration",
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"text_description": "Segment Area Formula (radians)\nA segment is the part of a circle between a chord and its arc.\nUse the formula below whenever the central angle is measured in radians.\nKey Points:\nArea \\(= \\dfrac{1}{2} r^{2} (\\theta - \\sin \\theta)\\)\n\\(\\theta\\) is the central angle in radians.\nConvert degrees to radians before applying the formula.",
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}
]
},
{
"slide": 8,
"fragments": [
{
"fragment_index": -1,
"text_description": "Multiple Choice Question\nconst correctOption = 1;\n const answerCards = document.querySelectorAll('.answer-card');\n const submitBtn = document.getElementById('slide-09-a3c5f2-submitBtn');\n const feedbackCorrect = document.getElementById('slide-09-a3c5f2-feedbackCorrect');\n const feedbackIncorrect = document.getElementById('slide-09-a3c5f2-feedbackIncorrect');\n\n let selectedOption = null;\n\n answerCards.forEach((card, index) => {\n card.addEventListener('click', () => {\n answerCards.forEach(c => c.classList.remove('border-blue-500', 'bg-blue-50'));\n card.classList.add('border-blue-500', 'bg-blue-50');\n selectedOption = index;\n });\n });\n\n submitBtn.addEventListener('click', () => {\n if (selectedOption === null) return;\n\n if (selectedOption === correctOption) {\n feedbackCorrect.classList.remove('hidden');\n feedbackIncorrect.classList.add('hidden');\n } else {\n feedbackIncorrect.classList.remove('hidden');\n feedbackCorrect.classList.add('hidden');\n }\n });",
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},
{
"fragment_index": 1,
"text_description": "Question\nθ = 270°. Which sector of the circle is larger?",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "1\nMinor sector",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "2\nMajor sector",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Hint:\nA minor sector spans less than 180°.",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "Submit Answer",
"image_description": ""
},
{
"fragment_index": 6,
"text_description": "Correct!\nYes. 270° is more than half a circle, so it forms the major sector.",
"image_description": ""
},
{
"fragment_index": 7,
"text_description": "Incorrect\nCheck again: a minor sector is ≤ 180°, so 270° cannot be minor.",
"image_description": ""
}
]
},
{
"slide": 9,
"fragments": [
{
"fragment_index": -1,
"text_description": "Alternate Segment",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Angle between tangent and chord equals angle in opposite arc.",
"image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/presentation_images/s3.amazonaws.com/media-p.slid.es/uploads/2766134/images/11662301/pasted-from-clipboard.png"
},
{
"fragment_index": 2,
"text_description": "Alternate Segment Theorem\nIn circle theorems, this rule links a tangent–chord angle to an angle inside the circle.\nRecognising this pair allows you to find unknown angles quickly.\nKey Points:\nAngle between a tangent and its chord at the point of contact.\nEquals the angle the same chord subtends in the alternate (opposite) segment.\nUse it to spot equal angles when solving circle problems.",
"image_description": ""
}
]
},
{
"slide": 10,
"fragments": [
{
"fragment_index": -1,
"text_description": "Angle at Centre",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Central angle is twice the corresponding circumferential angle.",
"image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/presentation_images/asset.sparkl.me/pb/presentation/2533/images/bc3f1d9006c0304d1e7e17e5a3eeff65.png"
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"fragment_index": 2,
"text_description": "Centre Angle is Double\nIn circle geometry, the angle at the centre is special.\nIt is always twice the angle at the circumference on the same arc.\nKey Points:\n\\( \\angle AOB = 2 \\times \\angle ACB \\)\nBoth angles subtend the same chord or arc AB.\nUse this circle theorem to find unknown angles quickly.",
"image_description": ""
}
]
},
{
"slide": 11,
"fragments": [
{
"fragment_index": -1,
"text_description": "Same Segment Angles",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Angles in the same segment are equal.",
"image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/presentation_images/s3.amazonaws.com/media-p.slid.es/uploads/2766134/images/11662301/pasted-from-clipboard.png"
},
{
"fragment_index": 2,
"text_description": "Circle Theorem\nCircle theorem: Angles in the same segment are equal.\nAny two angles subtended by the same chord on the same side share the same measure.\nKey Points:\nApplies to any chord of a circle.\nBoth angles lie on the circumference, same side of the chord.\nHandy for proving equal angles in geometric proofs.",
"image_description": ""
}
]
},
{
"slide": 12,
"fragments": [
{
"fragment_index": -1,
"text_description": "Semicircle Right Angle",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "The angle in a semicircle is always 90°.",
"image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/presentation_images/asset.sparkl.me/pb/presentation/2614/images/d6580c5b7c37743fce80472ed5ce8bf6.png"
},
{
"fragment_index": 2,
"text_description": "Diameter creates a right angle at the circumference\nIn any circle, a diameter subtends a right angle at every point on the circumference.\nThis circle theorem is often called the “angle in a semicircle” rule.\nKey Points:\nEndpoints of the diameter act as a chord.\nAny point on the semicircle forms a 90° angle.\nUseful for spotting right triangles in circle problems.",
"image_description": ""
}
]
},
{
"slide": 13,
"fragments": [
{
"fragment_index": -1,
"text_description": "Perpendicular Bisector",
"image_description": ""
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{
"fragment_index": 1,
"text_description": "",
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"text_description": "Circle Theorem: Perpendicular from Centre Bisects Chord\nDraw radius \\(OX\\) so that \\(OX \\perp AB\\). \\(O\\) is the centre of the circle.\nRight-angled triangles \\(\\triangle OAX\\) and \\(\\triangle OBX\\) are congruent because \\(OA = OB\\) and \\(OX\\) is common.\nTherefore \\(AX = XB\\). The perpendicular from the centre splits the chord equally, confirming the theorem.\nKey Points:\nLine starts at circle centre \\(O\\).\nMeets chord at 90°.\nForms two congruent right-angled triangles.\nHence, chord is bisected: \\(AX = XB\\).",
"image_description": ""
}
]
},
{
"slide": 14,
"fragments": [
{
"fragment_index": -1,
"text_description": "Cyclic Quadrilateral",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Opposite angles A & C and B & D form straight lines.",
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{
"fragment_index": 2,
"text_description": "Opposite Angles Are Supplementary\nA quadrilateral with all four vertices on one circle is called cyclic.\nCircle theorem: each pair of opposite interior angles sums to \\(180^{\\circ}\\).",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Key Points:\nAll four vertices lie on the same circle.\n\\(∠A + ∠C = 180^{\\circ}\\).\n\\(∠B + ∠D = 180^{\\circ}\\).",
"image_description": ""
}
]
},
{
"slide": 15,
"fragments": [
{
"fragment_index": -1,
"text_description": "Tangent Facts",
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"text_description": "",
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"fragment_index": 2,
"text_description": "Two Essential Tangent Properties\nAfter this slide, you should be able to state both key tangent properties clearly.",
"image_description": ""
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{
"fragment_index": 3,
"text_description": "Key Points:\nThe tangent is perpendicular to the radius at the point of contact.\nTangents drawn from the same external point are equal in length.",
"image_description": ""
}
]
},
{
"slide": 16,
"fragments": [
{
"fragment_index": -1,
"text_description": "Tangent–Secant Rule",
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},
{
"fragment_index": 1,
"text_description": "Tangent PA and secant PBC from point P",
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{
"fragment_index": 2,
"text_description": "Power of a Point\nFrom external point P, draw tangent PA and secant PBC to the circle.\nPower of a Point states the square of the tangent equals the external part times the whole secant.",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "Key Points:\nRule: \\(PA^{2}=PB\\cdot PC\\)\nUse it to find any missing length when two are known.\nValid for every circle sharing the same external point.",
"image_description": ""
}
]
},
{
"slide": 17,
"fragments": [
{
"fragment_index": -1,
"text_description": "Secant–Secant Rule",
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},
{
"fragment_index": 1,
"text_description": "Intersecting secants from point P",
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{
"fragment_index": 2,
"text_description": "Power of a Point result\nFrom an outside point P, draw two secants \\(PAB\\) and \\(PCD\\) to the circle.\nThe Power of a Point theorem states the products of their parts are equal.\nKey Points:\n\\(PA \\cdot PB = PC \\cdot PD\\)\nProduct of near and far parts of each secant is equal.\nQuickly find unknown lengths using the relation.",
"image_description": ""
}
]
},
{
"slide": 18,
"fragments": [
{
"fragment_index": -1,
"text_description": "Key Takeaways\nRecap of essential facts about circles.",
"image_description": ""
},
{
"fragment_index": 1,
"text_description": "Parts & Symbols\nRadius, diameter, chord and arc describe circle parts; central angle θ locates the arc.",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "Key Formulas\nArc length \\(L = 2\\pi r \\frac{\\theta}{360^\\circ}\\); sector area \\(A = \\frac12 r^2 \\theta\\) (θ in radians).",
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},
{
"fragment_index": 3,
"text_description": "Angle Theorems\nAngle at centre = 2 × inscribed angle; same-segment angles equal; semicircle angle is 90°.",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "Tangents & Secants\nTangent ⟂ radius; equal tangents from one point; tangent-secant and secant-secant products relate lengths.",
"image_description": ""
}
]
},
{
"slide": 19,
"fragments": [
{
"fragment_index": -1,
"text_description": "Multiple Choice Question\nconst correctOption = 2;\n const answerCards = document.querySelectorAll('.answer-card');\n const submitBtn = document.getElementById('submitBtn');\n const feedbackCorrect = document.getElementById('feedbackCorrect');\n const feedbackIncorrect = document.getElementById('feedbackIncorrect');\n\n let selectedOption = null;\n\n answerCards.forEach((card, index) => {\n card.addEventListener('click', () => {\n answerCards.forEach(c => c.classList.remove('border-blue-500','bg-blue-50'));\n card.classList.add('border-blue-500','bg-blue-50');\n selectedOption = index;\n });\n });\n\n submitBtn.addEventListener('click', () => {\n if (selectedOption === null) return;\n\n if (selectedOption === correctOption) {\n feedbackCorrect.classList.remove('hidden');\n feedbackIncorrect.classList.add('hidden');\n } else {\n feedbackIncorrect.classList.remove('hidden');\n feedbackCorrect.classList.add('hidden');\n }\n });",
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"fragment_index": 1,
"text_description": "Question\nA chord subtends 50° at the circumference. What angle does the same chord subtend at the centre?",
"image_description": ""
},
{
"fragment_index": 2,
"text_description": "1\n25°",
"image_description": ""
},
{
"fragment_index": 3,
"text_description": "2\n50°",
"image_description": ""
},
{
"fragment_index": 4,
"text_description": "3\n100°",
"image_description": ""
},
{
"fragment_index": 5,
"text_description": "4\n150°",
"image_description": ""
},
{
"fragment_index": 6,
"text_description": "Hint:\nRecall: the angle at the centre is twice the angle at the circumference for the same chord.",
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{
"fragment_index": 7,
"text_description": "Submit Answer",
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{
"fragment_index": 8,
"text_description": "Correct!\nGreat job! By the Angle at the Centre Theorem, 2 × 50° = 100°.",
"image_description": ""
},
{
"fragment_index": 9,
"text_description": "Incorrect\nReview the Angle at the Centre Theorem: central angle = 2 × angle at circumference.",
"image_description": ""
}
]
}
]