{
  "title": "Equal Chords and Equal Distances - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "If two chords are equal, their distances from center are:", "options": {"A": "Equal", "B": "Unequal", "C": "Inversely proportional", "D": "Directly proportional"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Conversely, if distances from center to two chords are equal, chords are:", "options": {"A": "Equal", "B": "Parallel", "C": "Perpendicular", "D": "Of different lengths"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "In a circle, the longest chord is:", "options": {"A": "Closest to center", "B": "Farthest from center", "C": "At medium distance", "D": "Diameter"}, "correct_answer": "D"},
    {"id": 4, "difficulty": "easy", "question": "If two chords are at equal distances from center, they subtend _____ angles at center.", "options": {"A": "Equal", "B": "Unequal", "C": "Complementary", "D": "Supplementary"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "In circle radius 10cm, chord length 16cm. Another equal chord will be at distance:", "options": {"A": "5cm", "B": "6cm", "C": "8cm", "D": "10cm"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "Two equal chords always make equal angles at:", "options": {"A": "Any point on circle", "B": "Center", "C": "Circumference", "D": "All of these"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "If chord length is doubled, distance from center:", "options": {"A": "Doubles", "B": "Halves", "C": "Changes non-linearly", "D": "Remains same"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "In circle, chords equidistant from center and on same side of center are:", "options": {"A": "Parallel", "B": "Equal", "C": "Congruent", "D": "B and C"}, "correct_answer": "D"},
    {"id": 9, "difficulty": "easy", "question": "If two chords are parallel, they are:", "options": {"A": "Always equal", "B": "Always equidistant from center", "C": "Sometimes equal", "D": "Never equal"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "The perpendicular from center to chord:", "options": {"A": "Bisects the chord", "B": "Bisects the arc", "C": "Both A and B", "D": "Neither"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "Equal chords cut off equal:", "options": {"A": "Arcs", "B": "Segments", "C": "Sectors", "D": "All of these"}, "correct_answer": "D"},
    {"id": 12, "difficulty": "easy", "question": "If two chords have equal distances from center but are on opposite sides, they are:", "options": {"A": "Equal", "B": "Parallel", "C": "Perpendicular", "D": "None of these"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Chord nearer to center is _____ than chord farther away.", "options": {"A": "Shorter", "B": "Longer", "C": "Equal", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "In two concentric circles, chord of larger circle touching smaller circle is _____ chord of smaller circle.", "options": {"A": "Equal to", "B": "Longer than", "C": "Diameter of", "D": "Tangent to"}, "correct_answer": "C"},
    {"id": 15, "difficulty": "easy", "question": "If distance from center to chord is zero, chord is:", "options": {"A": "Minimum length", "B": "Maximum length", "C": "Zero length", "D": "Normal length"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "As chord moves away from center, its length:", "options": {"A": "Increases", "B": "Decreases", "C": "First increases then decreases", "D": "Remains constant"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "Equal chords in congruent circles:", "options": {"A": "Are equidistant from centers", "B": "Subtend equal angles", "C": "Both A and B", "D": "Neither"}, "correct_answer": "C"},
    {"id": 18, "difficulty": "easy", "question": "If two chords subtend equal angles at center, they are:", "options": {"A": "Equal", "B": "Parallel", "C": "Perpendicular", "D": "None of these"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "The locus of midpoints of equal chords parallel to given direction is:", "options": {"A": "Diameter perpendicular to direction", "B": "Line through center", "C": "Circle", "D": "None of these"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "If chords AB and CD are equal, and OM ⟂ AB, ON ⟂ CD, then:", "options": {"A": "OM = ON", "B": "AM = CN", "C": "Both A and B", "D": "Neither"}, "correct_answer": "C"},
   {"id": 21, "difficulty": "medium", "question": "In circle radius 13cm, two equal chords lengths 24cm each are on same side of center. Distance between chords if they are parallel?", "options": {"A": "0 cm", "B": "5cm", "C": "7cm", "D": "10cm"}, "correct_answer": "A"},  {"id": 22, "difficulty": "medium", "question": "Two equal chords intersect at right angles. If each chord length is 10√2 cm, distance from intersection to center is 5cm. Radius?", "options": {"A": "10cm", "B": "12cm", "C": "13cm", "D": "15cm"}, "correct_answer": "A"},
   {"id": 23, "difficulty": "medium", "question": "Chord length 30cm at distance 8cm from center. Another chord of same circle length 34cm, its distance from center?", "options": {"A": "0 cm", "B": "√15 cm", "C": "√17 cm", "D": "√21 cm"}, "correct_answer": "A"}, {"id": 24, "difficulty": "medium", "question": "Two parallel chords lengths 16cm and 30cm on same side of center, distance between chords 7cm. Radius?", "options": {"A": "17cm", "B": "18cm", "C": "19cm", "D": "20cm"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Equal chords AB and AC contain angle 60° at A. If radius is 10cm, distance from center to BC?", "options": {"A": "5cm", "B": "5√3 cm", "C": "10cm", "D": "10√3 cm"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Two equal chords subtend angles 60° and 120° at center. Ratio of distances from center?", "options": {"A": "1:√3", "B": "√3:1", "C": "1:2", "D": "2:1"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "In circle radius 25cm, chord length 14cm. Another chord equal to first, distance between their midpoints if chords are parallel?", "options": {"A": "√2304 cm", "B": "48cm", "C": "24cm", "D": "12cm"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "Two equal chords intersect inside circle. Segments of one chord are 6cm and 4cm. Length of each chord?", "options": {"A": "10cm", "B": "12cm", "C": "14cm", "D": "16cm"}, "correct_answer": "A"},
   {"id": 29, "difficulty": "medium", "question": "Chord length equal to radius subtends angle θ at center. Another chord at distance r/2 from center subtends angle φ. Ratio θ:φ?", "options": {"A": "1:2", "B": "2:1", "C": "√3:1", "D": "1:√3"}, "correct_answer": "A"}, {"id": 30, "difficulty": "medium", "question": "Two concentric circles radii 5cm and 13cm. Chord of larger circle touches smaller circle. If this chord is equal to chord of smaller circle, chord length?", "options": {"A": "12cm", "B": "18cm", "C": "24cm", "D": "26cm"}, "correct_answer": "C"},
    {
      "id": 31,
      "difficulty": "medium",
      "question": "Through point P inside circle, two equal chords PA and PB drawn. If ∠APB=60° and OP=6cm, radius 10cm, length of each chord?",
      "options": {"A": "8√3 cm", "B": "12 cm", "C": "10√2 cm", "D": "16 cm"},
      "correct_answer": "B"
    },       {
      "id": 32,
      "difficulty": "medium",
      "question": "Two parallel equal chords on opposite sides of center are at distances 12cm and 5cm from center. Find radius of circle.",
      "options": {"A": "13cm", "B": "15cm", "C": "17cm", "D": "√119 cm"},
      "correct_answer": "A"
    },    {"id": 33, "difficulty": "medium", "question": "Regular hexagon inscribed in circle radius R. Distance from center to side?", "options": {"A": "R√3/2", "B": "R/2", "C": "R√2/2", "D": "R/√2"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Two equal chords AB and CD intersect at E. If AE=8, EB=2, and CE=4, length of each chord?", "options": {"A": "10", "B": "12", "C": "14", "D": "16"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Chord length 2R sin(θ/2). For two equal chords with θ1 and θ2, ratio of distances from center?", "options": {"A": "cos(θ1/2):cos(θ2/2)", "B": "sin(θ1/2):sin(θ2/2)", "C": "tan(θ1/2):tan(θ2/2)", "D": "cot(θ1/2):cot(θ2/2)"}, "correct_answer": "A"},
   {"id": 36, "difficulty": "medium", "question": "In circle radius 10cm, two parallel equal chords each length 12cm on opposite sides of center distance 16cm apart. Possible?", "options": {"A": "Yes", "B": "No", "C": "Only if chords perpendicular", "D": "Only if chords diameters"}, "correct_answer": "A"}, {"id": 37, "difficulty": "medium", "question": "Point P inside circle such that two equal chords through P have lengths 20cm each. Shortest chord through P has length 12cm. Distance OP?", "options": {"A": "8cm", "B": "10cm", "C": "12cm", "D": "14cm"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Two equal chords subtend angles 90° and 60° at center. Ratio of their distances from center?", "options": {"A": "√2:√3", "B": "√3:√2", "C": "1:√2", "D": "1:√3"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Chord length 24cm at distance 5cm from center. Another chord equal to first, at what distance if angle between them at center is 90°?", "options": {"A": "5cm", "B": "10cm", "C": "12cm", "D": "13cm"}, "correct_answer": "A"},
   {"id": 40, "difficulty": "medium", "question": "Two concentric circles, chord of larger equal to diameter of smaller. If radii are R and r with R>r, relation?", "options": {"A": "R=√2 r", "B": "R=2r", "C": "R=√3 r", "D": "R=√5 r"}, "correct_answer": "A"},{"id": 41, "difficulty": "medium", "question": "Equal chords AB and AC contain angle 120° at A. Triangle ABC is:", "options": {"A": "Isosceles", "B": "Equilateral", "C": "Right", "D": "Obtuse"}, "correct_answer": "A"},  {"id": 42, "difficulty": "medium", "question": "Through point distance d from center, two equal chords drawn making angle 2α with each other. Length of each chord?", "options": {"A": "2√(R²-d² sin²α)", "B": "2√(R²-d² cos²α)", "C": "2√(R²-d² tan²α)", "D": "2√(R²-d² cot²α)"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "In two congruent circles, equal chords subtend angles 60° and 120° at respective centers. Ratio of distances from centers?", "options": {"A": "1:2", "B": "√3:1", "C": "2:1", "D": "1:√3"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "Two equal chords intersect at right angles. Distance from intersection to center equals half radius. Angle subtended by chord at center?", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "medium", "question": "Regular pentagon inscribed in circle radius R. Distance from center to side?", "options": {"A": "R cos 36°", "B": "R sin 36°", "C": "R cos 54°", "D": "R sin 54°"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "Chord length equal to side of inscribed equilateral triangle. Its distance from center?", "options": {"A": "R/2", "B": "R√3/2", "C": "R/√2", "D": "R√2/2"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "Two equal chords of circle with center O intersect at P. If OP is perpendicular bisector of one chord, then chords are:", "options": {"A": "Perpendicular", "B": "Make 60°", "C": "Make 45°", "D": "Parallel"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "From point on circle, two equal chords drawn to endpoints of diameter. Angle between chords?", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "D"},
    {"id": 49, "difficulty": "medium", "question": "In circle radius 5cm, two equal chords lengths 6cm each. Maximum possible distance between their midpoints?", "options": {"A": "8cm", "B": "10cm", "C": "12cm", "D": "14cm"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "Two equal chords intersect such that segments of one are 9cm and 4cm. Length of each chord?", "options": {"A": "13cm", "B": "15cm", "C": "17cm", "D": "19cm"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "Chord length 2R sin 18°. Another chord equal to first at distance R cos 36° from center. True?", "options": {"A": "Yes", "B": "No", "C": "Only for specific R", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "Two parallel equal chords on same side of center distance x apart. If each chord length 2l and radius R, then x=?", "options": {"A": "2√(R²-l²)", "B": "√(R²-l²)", "C": "0", "D": "2l"}, "correct_answer": "C"},
    {"id": 53, "difficulty": "medium", "question": "Through point P inside circle, shortest chord has length 8cm and longest chord 20cm. Two equal chords through P making angle 60° have length?", "options": {"A": "√84 cm", "B": "√96 cm", "C": "√108 cm", "D": "√120 cm"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "Equal chords in two concentric circles. If radii are R and r, ratio of chord lengths?", "options": {"A": "R:r", "B": "√(R²-d²):√(r²-d²)", "C": "R²:r²", "D": "√R:√r"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "medium", "question": "Regular octagon inscribed in circle. Distance from center to side?", "options": {"A": "R cos 22.5°", "B": "R sin 22.5°", "C": "R/√2", "D": "R√2/2"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Two equal chords AB and CD intersect at E inside circle. If AE=12, BE=3, CE=4, find ED.", "options": {"A": "9", "B": "12", "C": "16", "D": "20"}, "correct_answer": "A"},
{"id": 57, "difficulty": "medium", "question": "Chord length equal to distance from center. Angle subtended at center by chord?", "options": {"A": "2 arcsin(1/√5)", "B": "2 arccos(1/√5)", "C": "120°", "D": "90°"}, "correct_answer": "A"}, {"id": 57, "difficulty": "medium", "question": "Chord length equal to distance from center. Angle subtended at center by chord?", "options": {"A": "2 arcsin(√5-1)/2", "B": "2 arccos(√5-1)/2", "C": "120°", "D": "90°"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Two equal chords on opposite sides of center distance 17cm apart. If radius 13cm, chord length?", "options": {"A": "10cm", "B": "12cm", "C": "20cm", "D": "24cm"}, "correct_answer": "D"},
    {"id": 59, "difficulty": "medium", "question": "From point distance 5cm from center, two equal chords drawn making angle 120°. If radius 13cm, chord length?", "options": {"A": "12√3 cm", "B": "24cm", "C": "20cm", "D": "18cm"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "Regular decagon inscribed in circle. Distance from center to side?", "options": {"A": "R cos 18°", "B": "R sin 18°", "C": "R cos 36°", "D": "R sin 36°"}, "correct_answer": "A"},
    {
      "id": 61,
      "difficulty": "hard",
      "question": "Two equal chords intersect at right angles. Segments of one chord are a and b. Radius in terms of a,b?",
      "options": {"A": "√((a²+b²)(a+b)²/(4ab))", "B": "√((a²+b²)(a+b)²/(2ab))", "C": "√((a+b)²(a²+b²)/(4ab))", "D": "√((a+b)²(a²+b²)/(2ab))"},
      "correct_answer": "C"
    },    {"id": 62, "difficulty": "hard", "question": "Through fixed point inside circle, chords drawn. Product of lengths of segments of chords through point constant. For equal chords making angle θ, product?", "options": {"A": "R²-d²", "B": "(R²-d²) sin²(θ/2)", "C": "(R²-d²) cos²(θ/2)", "D": "(R²-d²) tan²(θ/2)"}, "correct_answer": "C"},
    {"id": 63, "difficulty": "hard", "question": "Two equal chords AB and AC contain angle 2α at A. Distance from center to BC?", "options": {"A": "R cos α", "B": "R sin α", "C": "R tan α", "D": "R cot α"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "hard", "question": "From point on circle, two equal chords drawn to divide circle into three equal arcs. Angle between chords?", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "hard", "question": "Three equal chords of circle divide circumference into three arcs ratio 1:2:3. Angles subtended by chords at center?", "options": {"A": "60°,120°,180°", "B": "60°,90°,210°", "C": "60°,100°,200°", "D": "Not possible"}, "correct_answer": "D"},
    {"id": 66, "difficulty": "hard", "question": "Two equal chords intersect at P. Line through P and center O meets circle at Q,R. Then P is midpoint of QR if:", "options": {"A": "Chords perpendicular", "B": "Chords equal", "C": "OP=0", "D": "Always"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "hard", "question": "Regular polygon n sides inscribed. Distance from center to side R cos(π/n). For n=24, distance approximately?", "options": {"A": "0.991R", "B": "0.966R", "C": "0.924R", "D": "0.866R"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "hard", "question": "Two equal chords in circle x²+y²=25 have midpoints lying on line 3x+4y=0. Length of each chord?", "options": {"A": "6", "B": "8", "C": "10", "D": "12"}, "correct_answer": "B"},
    {"id": 69, "difficulty": "hard", "question": "From point P, two equal chords PA and PB drawn. If PA=PB=l and ∠APB=θ, distance OP=?", "options": {"A": "√(R²-l²/4 sin²(θ/2))", "B": "√(R²-l²/4 cos²(θ/2))", "C": "√(R²-l² sin²(θ/2))", "D": "√(R²-l² cos²(θ/2))"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "hard", "question": "Chord length equal to side of inscribed regular n-gon. As n→∞, distance from center approaches:", "options": {"A": "0", "B": "R", "C": "R/2", "D": "2R"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "hard", "question": "Two equal chords AB and CD intersect at E. Circles on AE and CE as diameters intersect at E and F. Then F lies on:", "options": {"A": "Chord BD", "B": "Line through E perpendicular to AC", "C": "Original circle", "D": "None"}, "correct_answer": "C"},
    {"id": 72, "difficulty": "hard", "question": "Three equal chords divide circle into four regions. If chords concurrent, minimum possible area of central region?", "options": {"A": "R²(π/3-√3/2)", "B": "R²(π/2-1)", "C": "R²(π/4-1/2)", "D": "0"}, "correct_answer": "D"},
    {"id": 73, "difficulty": "hard", "question": "From point P, two equal chords drawn making angles α,β with line OP. If chords equal, relation between α,β?", "options": {"A": "α=β", "B": "α+β=90°", "C": "α+β=180°", "D": "No relation"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "hard", "question": "Two equal chords subtend angles 2α and 2β at center. Distance between chords if on same side?", "options": {"A": "R(cos α - cos β)", "B": "R(sin α - sin β)", "C": "R(tan α - tan β)", "D": "R(cot α - cot β)"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "hard", "question": "Chord length 2R sin(π/2n) in regular n-gon. For n=5, distance from center?", "options": {"A": "R cos(π/5)", "B": "R sin(π/5)", "C": "R cos(π/10)", "D": "R sin(π/10)"}, "correct_answer": "A"},
       {
      "id": 76,
      "difficulty": "hard", 
      "question": "Two equal chords intersect at right angles. Ratio of segments of one chord is 3:4. Find ratio of radius to chord length.",
      "options": {"A": "5:7", "B": "25:48", "C": "5:4√2", "D": "√58:7"},
      "correct_answer": "D"
    },    {"id": 77, "difficulty": "hard", "question": "Through point distance d from center, chord drawn making angle θ with line through center. Length?", "options": {"A": "2√(R²-d² sin²θ)", "B": "2√(R²-d² cos²θ)", "C": "2√(R²-d² tan²θ)", "D": "2√(R²-d² cot²θ)"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "hard", "question": "Two equal chords intersect. Line joining intersection to center bisects angle between chords if:", "options": {"A": "Chords equal", "B": "Chords perpendicular", "C": "Always", "D": "Never"}, "correct_answer": "C"},
    {"id": 79, "difficulty": "hard", "question": "From point on circle, two equal chords drawn to divide circle into arcs ratio 1:2:1:2 (alternating). Angle between chords?", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "B"},
    {"id": 80, "difficulty": "hard", "question": "Regular polygon inscribed. Ratio of distances from center to side for n and 2n?", "options": {"A": "cos(π/2n):cos(π/n)", "B": "cos(π/n):cos(π/2n)", "C": "sin(π/2n):sin(π/n)", "D": "tan(π/2n):tan(π/n)"}, "correct_answer": "B"},
    {"id": 81, "difficulty": "hard", "question": "Two equal chords AB, AC with ∠BAC=60°. Area triangle ABC maximum when?", "options": {"A": "AB diameter", "B": "AB=AC=R", "C": "AB=AC=R√3", "D": "Always same"}, "correct_answer": "B"},
    {"id": 82, "difficulty": "hard", "question": "Chord length equal to distance from center. As chord rotates, midpoint traces:", "options": {"A": "Circle", "B": "Line", "C": "Ellipse", "D": "Parabola"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Two equal chords intersect. Lines joining endpoints form quadrilateral. It is cyclic if:", "options": {"A": "Chords perpendicular", "B": "Chords equal", "C": "Always", "D": "Never"}, "correct_answer": "C"},
    {"id": 84, "difficulty": "hard", "question": "Three equal chords divide circle into 6 arcs. If arcs alternate between α and β with α+β=120°, chords concurrent?", "options": {"A": "Yes", "B": "No", "C": "Only if α=β", "D": "Only if α=60°"}, "correct_answer": "A"},
        {
      "id": 85,
      "difficulty": "hard",
      "question": "From point P inside circle, two equal chords PA and PB. If ∠APB=90° and OP=5cm, radius 13cm, find area of triangle APB.",
      "options": {"A": "60 cm²", "B": "72 cm²", "C": "84 cm²", "D": "96 cm²"},
      "correct_answer": "A"
    },    {"id": 86, "difficulty": "hard", "question": "Two equal chords subtend angles 90° and 60° at point on circumference. Ratio of distances from center?", "options": {"A": "√2:√3", "B": "√3:√2", "C": "1:√3", "D": "√3:1"}, "correct_answer": "B"},
    {"id": 87, "difficulty": "hard", "question": "Chord length 2R sin θ. For two equal chords with angles θ,φ at center, sum of distances from center?", "options": {"A": "R(cos θ+cos φ)", "B": "R(sin θ+sin φ)", "C": "R(tan θ+tan φ)", "D": "R(cot θ+cot φ)"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Regular n-gon inscribed. As n increases, distance from center to side:", "options": {"A": "Increases to R", "B": "Decreases to 0", "C": "Increases to R/2", "D": "Decreases to R/2"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Two equal chords intersect. Lines joining their endpoints form two triangles. These triangles are:", "options": {"A": "Similar", "B": "Congruent", "C": "Equal area", "D": "All of these"}, "correct_answer": "D"},
    {"id": 90, "difficulty": "hard", "question": "From point distance d from center, chord drawn. Midpoint traces circle radius √(R²-d²)/2. For equal chords making angle 2α, distance between midpoints?", "options": {"A": "√(R²-d²) sin α", "B": "√(R²-d²) cos α", "C": "√(R²-d²) tan α", "D": "√(R²-d²) cot α"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "Three equal chords divide circle into 4 regions of equal area if:", "options": {"A": "They form equilateral triangle", "B": "They are diameters", "C": "They concur at center", "D": "Impossible"}, "correct_answer": "D"},
    {"id": 92, "difficulty": "hard", "question": "Chord length 2√(R²-d²). Two equal chords with distances d1,d2. If chords perpendicular, d1²+d2²=?", "options": {"A": "R²", "B": "2R²", "C": "R²/2", "D": "√2 R²"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "Two equal chords AB, AC. Circle with diameter BC meets AB at D. Then AD=?", "options": {"A": "AB/2", "B": "AB/3", "C": "AB/√2", "D": "AB/√3"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Regular polygon inscribed. Distance from center to side R cos(π/n). For n=7, approximate?", "options": {"A": "0.901R", "B": "0.875R", "C": "0.822R", "D": "0.782R"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Two equal chords intersect at P. Line through P perpendicular to one chord bisects other chord if:", "options": {"A": "Chords equal", "B": "P is midpoint", "C": "Always", "D": "Never"}, "correct_answer": "C"},
    {"id": 96, "difficulty": "hard", "question": "Chord length equal to side of inscribed square. Another chord equal to side of inscribed equilateral triangle. Ratio of distances from center?", "options": {"A": "√2:√3", "B": "√3:√2", "C": "1:√2", "D": "1:√3"}, "correct_answer": "B"},
    {"id": 97, "difficulty": "hard", "question": "From point on circle, two equal chords divide circumference into arcs ratio 2:3:2:3. Angle between chords?", "options": {"A": "72°", "B": "90°", "C": "108°", "D": "120°"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Two equal chords AB, CD intersect at E. If circles on AB and CD as diameters intersect at E and F, then EF is:", "options": {"A": "Perpendicular to line joining centers", "B": "Parallel to AC", "C": "Diameter of original circle", "D": "None"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Regular n-gon inscribed. Distance from center to side for n=10?", "options": {"A": "R cos 18°", "B": "R sin 18°", "C": "R cos 36°", "D": "R sin 36°"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Two equal chords intersect. Lines joining their endpoints form kite. Diagonals of kite intersect at:", "options": {"A": "Center of circle", "B": "Point of intersection of chords", "C": "Midpoint of chords", "D": "None"}, "correct_answer": "B"}
  ]
}