{
  "title": "Construction of circumcircle and incircle - Grade 10 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "📐 Which triangle has both circumcircle and incircle? ⭕", "options": {"A": "Only right triangle", "B": "Only equilateral triangle", "C": "Every triangle", "D": "Only isosceles triangle"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "🔺 To construct circumcircle, we need to find: 🔍", "options": {"A": "Incenter", "B": "Centroid", "C": "Circumcenter", "D": "Orthocenter"}, "correct_answer": "C"},
    {"id": 3, "difficulty": "easy", "question": "✏️ The incircle touches triangle's: 🤔", "options": {"A": "Vertices", "B": "Midpoints", "C": "All three sides", "D": "Only base"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "🎯 Circumcenter is intersection of: ✂️", "options": {"A": "Medians", "B": "Angle bisectors", "C": "Perpendicular bisectors", "D": "Altitudes"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "💫 Incenter is intersection of: ⚡", "options": {"A": "Perpendicular bisectors", "B": "Angle bisectors", "C": "Medians", "D": "Altitudes"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "📏 For incircle construction, we draw: 🎨", "options": {"A": "Perpendicular from incenter to side", "B": "Parallel lines", "C": "Angle bisectors only", "D": "Medians only"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "🔵 Circumcircle passes through: 🎯", "options": {"A": "Midpoints of sides", "B": "All three vertices", "C": "Points on sides", "D": "Foot of altitudes"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "🟢 Incircle is ______ to all sides: 🤝", "options": {"A": "Parallel", "B": "Perpendicular", "C": "Tangent", "D": "Secant"}, "correct_answer": "C"},
    {"id": 9, "difficulty": "easy", "question": "📐 For right triangle, circumcenter lies: 📍", "options": {"A": "Inside triangle", "B": "At hypotenuse midpoint", "C": "Outside triangle", "D": "At vertex"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "🎨 To draw circumcircle, we need: 🧭", "options": {"A": "Compass only", "B": "Ruler only", "C": "Both compass & ruler", "D": "Protractor only"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "🔺 In obtuse triangle, circumcenter is: 🌓", "options": {"A": "Inside triangle", "B": "On triangle", "C": "Outside triangle", "D": "At vertex"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "💎 For equilateral triangle, incenter and circumcenter: ⭐", "options": {"A": "Are different", "B": "Coincide", "C": "Are perpendicular", "D": "Are parallel"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "📐 Circumradius is distance from circumcenter to: 📏", "options": {"A": "Incenter", "B": "Any vertex", "C": "Midpoint", "D": "Side"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "🟡 Inradius is distance from incenter to: 📍", "options": {"A": "Vertex", "B": "Any side", "C": "Circumcenter", "D": "Midpoint"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "🎯 First step for circumcircle: 1️⃣", "options": {"A": "Draw triangle", "B": "Find circumcenter", "C": "Draw circle", "D": "Measure sides"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "✏️ To find incenter, we draw at least: 🔢", "options": {"A": "1 angle bisector", "B": "2 angle bisectors", "C": "3 angle bisectors", "D": "Perpendicular bisectors"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "🔵 Circumcircle center is ______ from vertices: ⚖️", "options": {"A": "Different distances", "B": "Same distance", "C": "Twice distance", "D": "Half distance"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "🟢 Incircle is always ______ triangle: 📐", "options": {"A": "Outside", "B": "Inside", "C": "Passing through", "D": "Around"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "📏 For construction, we use compass to: 🧭", "options": {"A": "Measure angles", "B": "Draw arcs", "C": "Draw straight lines", "D": "Calculate area"}, "correct_answer": "B"},
    {"id": 20, "difficulty": "easy", "question": "🎨 Complete statement: Incircle is ______ circle in triangle: 💫", "options": {"A": "Largest", "B": "Smallest", "C": "Only", "D": "Largest possible"}, "correct_answer": "D"},
    
    {"id": 21, "difficulty": "medium", "question": "📐 Triangle ABC has AB=6cm, BC=8cm, CA=10cm. What type of triangle for circumcenter? 🔺", "options": {"A": "Inside triangle", "B": "On hypotenuse midpoint", "C": "Outside triangle", "D": "At vertex A"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "🔍 In ΔPQR, ∠P=60°, ∠Q=60°. Where is circumcenter? 🎯", "options": {"A": "Inside triangle", "B": "Outside triangle", "C": "At vertex", "D": "On side PQ"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "✏️ Construct circumcircle of triangle with sides 5cm, 6cm, 7cm. Circumradius is approx: 📏", "options": {"A": "3.5 cm", "B": "4.0 cm", "C": "4.5 cm", "D": "5.0 cm"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "🎨 In ΔXYZ, XY=8cm, YZ=15cm, XZ=17cm. Incenter lies: 📍", "options": {"A": "On hypotenuse", "B": "Inside triangle", "C": "Outside triangle", "D": "At right angle vertex"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "🔺 For triangle with angles 30°, 60°, 90°, circumcenter is: ⭐", "options": {"A": "At 90° vertex", "B": "Midpoint of hypotenuse", "C": "At 60° vertex", "D": "Inside triangle"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "📐 Triangle with sides 7cm, 24cm, 25cm has circumradius: 🔢", "options": {"A": "12.5 cm", "B": "13 cm", "C": "14 cm", "D": "15 cm"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "💎 In equilateral Δ of side 6cm, distance between incenter & vertex: 📏", "options": {"A": "2√3 cm", "B": "3 cm", "C": "√3 cm", "D": "3√3 cm"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "🎯 Construct incircle of triangle with sides 8cm, 15cm, 17cm. Inradius is: 📐", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "🔵 For acute triangle, circumcenter is always: 🌈", "options": {"A": "Inside", "B": "Outside", "C": "On triangle", "D": "At vertex"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "🟢 Triangle area=24cm², semi-perimeter=12cm. Inradius=? 📊", "options": {"A": "1 cm", "B": "2 cm", "C": "3 cm", "D": "4 cm"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "📏 In right Δ, hypotenuse=13cm, circumradius=? 🎯", "options": {"A": "6.5 cm", "B": "7 cm", "C": "7.5 cm", "D": "8 cm"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "✂️ To construct incircle, after finding incenter, next step: 🔄", "options": {"A": "Draw circle with radius to vertex", "B": "Draw perpendicular to any side", "C": "Draw angle bisectors", "D": "Measure all sides"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "🔺 Equilateral triangle side=8cm. Ratio of circumradius to inradius: ⚖️", "options": {"A": "1:2", "B": "2:1", "C": "3:1", "D": "4:1"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "🎨 In ΔABC, AB=AC=10cm, BC=12cm. Where is circumcenter? 📍", "options": {"A": "On BC", "B": "Inside triangle", "C": "On altitude from A", "D": "At A"}, "correct_answer": "C"},
    {"id": 35, "difficulty": "medium", "question": "💫 For triangle with vertices (0,0), (4,0), (0,3), circumcenter coordinates: 📊", "options": {"A": "(2,1.5)", "B": "(2,2)", "C": "(1.5,2)", "D": "(2,0)"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "📐 In ΔPQR, PQ=PR. Circumcenter lies on: 🔍", "options": {"A": "Median from P", "B": "Altitude from P", "C": "Angle bisector of ∠P", "D": "Perpendicular bisector of QR"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "🔵 Circumcircle of right triangle has diameter equal to: 📏", "options": {"A": "Shortest side", "B": "Longest side", "C": "Sum of sides", "D": "Difference of sides"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "🟢 In isosceles right triangle, incenter divides altitude in ratio: ⚡", "options": {"A": "1:1", "B": "1:√2", "C": "√2:1", "D": "2:1"}, "correct_answer": "C"},
    {"id": 39, "difficulty": "medium", "question": "🎯 Triangle sides 9cm, 12cm, 15cm. Area=54cm². Inradius=? 📊", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "✏️ For obtuse triangle, perpendicular bisectors intersect: 🌓", "options": {"A": "Inside triangle", "B": "Outside triangle", "C": "On longest side", "D": "At obtuse angle vertex"}, "correct_answer": "B"},
    
    {"id": 41, "difficulty": "hard", "question": "📐 ΔABC has vertices A(0,0), B(6,0), C(0,8). Circumcenter coordinates? 🎯", "options": {"A": "(3,4)", "B": "(4,3)", "C": "(3,3)", "D": "(4,4)"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "🔍 Triangle with sides 13cm, 14cm, 15cm. Circumradius (approx)? 📏", "options": {"A": "7.5 cm", "B": "8.1 cm", "C": "8.7 cm", "D": "9.2 cm"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "hard", "question": "💎 In ΔPQR, PQ=10cm, QR=17cm, PR=21cm. Find inradius (approx) if area=84cm²? 📊", "options": {"A": "3.0 cm", "B": "3.5 cm", "C": "4.0 cm", "D": "4.5 cm"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "hard", "question": "🎨 Equilateral triangle side=12cm. Distance between circumcenter and incenter? 🔢", "options": {"A": "2√3 cm", "B": "3√3 cm", "C": "4√3 cm", "D": "6√3 cm"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "hard", "question": "🔺 Triangle vertices (1,2), (3,4), (5,0). Circumcenter's x-coordinate? 📍", "options": {"A": "2.5", "B": "3.0", "C": "3.5", "D": "4.0"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "hard", "question": "📏 In ΔABC, AB=13, BC=14, CA=15. Altitude from A meets BC at D. If inradius=4, find AD? ⚡", "options": {"A": "10", "B": "11", "C": "12", "D": "13"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "hard", "question": "✏️ Triangle sides 8cm, 10cm, 12cm. Ratio of circumradius to inradius? ⚖️", "options": {"A": "2.5:1", "B": "3:1", "C": "3.5:1", "D": "4:1"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "🔵 In acute triangle, distance from circumcenter to side is: 🌈", "options": {"A": "R cos A", "B": "R sin A", "C": "R tan A", "D": "R sec A"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "🟢 Triangle area=30cm², perimeter=30cm. Inradius=? 📐", "options": {"A": "1 cm", "B": "2 cm", "C": "3 cm", "D": "4 cm"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "hard", "question": "🎯 In right triangle legs 6cm & 8cm. Distance between incenter and circumcenter? 📏", "options": {"A": "√5 cm", "B": "2√5 cm", "C": "3√5 cm", "D": "4√5 cm"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "💫 Triangle with sides 5k, 6k, 7k has inradius 4cm. Find k? 🔢", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "C"},
    {"id": 52, "difficulty": "hard", "question": "📐 For triangle with angles 45°, 60°, 75°, ratio of circumradius to inradius (approx)? ⭐", "options": {"A": "2.1", "B": "2.4", "C": "2.7", "D": "3.0"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "hard", "question": "🔍 In ΔABC, a=10, b=17, c=21. Circumradius R? (Use formula R=abc/4Δ, Δ=84) 📊", "options": {"A": "10.625", "B": "11.25", "C": "12.5", "D": "13.75"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "🎨 Equilateral triangle. Distance from vertex to incenter is 6cm. Side length? ✂️", "options": {"A": "6√3 cm", "B": "9√3 cm", "C": "12√3 cm", "D": "15√3 cm"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "🔺 Triangle vertices (0,0), (a,0), (0,b). Circumcenter is (a/2, b/2) for which triangle? 🤔", "options": {"A": "All triangles", "B": "Right triangles only", "C": "Isosceles triangles", "D": "Equilateral triangles"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "hard", "question": "📏 In isosceles triangle base=12cm, equal sides=10cm. Inradius? 📐", "options": {"A": "2.4 cm", "B": "2.8 cm", "C": "3.2 cm", "D": "3.6 cm"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "✏️ Triangle area=60cm², inradius=3cm. Semi-perimeter=? 📊", "options": {"A": "15 cm", "B": "18 cm", "C": "20 cm", "D": "24 cm"}, "correct_answer": "C"},
    {"id": 58, "difficulty": "hard", "question": "🔵 Circumcircle passes through triangle's vertices. Minimum arcs needed to construct? 🎯", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "hard", "question": "🟢 In triangle with sides 7, 24, 25, distance from incenter to right angle vertex? ⚡", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "💎 Triangle circumradius=10cm, one side=12cm. Angle opposite this side? 📏", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"}
  ]
}