{
  "title": "Median and Altitude (Triangles) - Grade 7 CBSE",
  "total_questions": 150,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "A median connects vertex to:", "options": {"A": "Opposite side's midpoint", "B": "Opposite vertex", "C": "Foot of perpendicular", "D": "Angle bisector"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "An altitude is ________ from vertex to opposite side.", "options": {"A": "Perpendicular", "B": "Parallel", "C": "Equal", "D": "Diagonal"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "How many medians does a triangle have?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "How many altitudes in a triangle?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "Medians intersect at:", "options": {"A": "Incenter", "B": "Centroid", "C": "Circumcenter", "D": "Orthocenter"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "Altitudes intersect at:", "options": {"A": "Centroid", "B": "Orthocenter", "C": "Incenter", "D": "Circumcenter"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "Centroid divides median in ratio:", "options": {"A": "1:1", "B": "1:2", "C": "2:1", "D": "3:1"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "In right triangle, altitude to hypotenuse creates:", "options": {"A": "Similar triangles", "B": "Congruent triangles", "C": "Equal triangles", "D": "Parallel lines"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "In equilateral triangle, median and altitude are:", "options": {"A": "Different", "B": "Same", "C": "Parallel", "D": "Perpendicular"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "Altitude can lie outside triangle in:", "options": {"A": "Acute triangle", "B": "Right triangle", "C": "Obtuse triangle", "D": "Equilateral triangle"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "Median divides triangle into:", "options": {"A": "Two equal areas", "B": "Two similar triangles", "C": "Two congruent triangles", "D": "Two right triangles"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "In isosceles triangle, altitude to base is also:", "options": {"A": "Median", "B": "Angle bisector", "C": "Both", "D": "Neither"}, "correct_answer": "C"},
    {"id": 13, "difficulty": "easy", "question": "If G is centroid and median length = 12 cm, distance from vertex to G = ?", "options": {"A": "4 cm", "B": "6 cm", "C": "8 cm", "D": "9 cm"}, "correct_answer": "C"},
    {"id": 14, "difficulty": "easy", "question": "Altitude is used to calculate:", "options": {"A": "Area", "B": "Perimeter", "C": "Volume", "D": "Angle"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "For obtuse triangle, orthocenter lies:", "options": {"A": "Inside", "B": "On triangle", "C": "Outside", "D": "At vertex"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "All three medians are equal in:", "options": {"A": "Right triangle", "B": "Isosceles triangle", "C": "Equilateral triangle", "D": "Scalene triangle"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "In right triangle, orthocenter is at:", "options": {"A": "Vertex of right angle", "B": "Midpoint of hypotenuse", "C": "Outside", "D": "Inside"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "If altitude = 8 cm and base = 10 cm, area = ?", "options": {"A": "40 cm²", "B": "48 cm²", "C": "64 cm²", "D": "80 cm²"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Centroid is also called:", "options": {"A": "Center of gravity", "B": "Center of circle", "C": "Center of symmetry", "D": "Center of rotation"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "In acute triangle, orthocenter lies:", "options": {"A": "Inside", "B": "Outside", "C": "On triangle", "D": "At vertex"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "In ΔABC, AD is median. If area ΔABD = 15 cm², area ΔADC = ?", "options": {"A": "10 cm²", "B": "15 cm²", "C": "20 cm²", "D": "30 cm²"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "In right ΔABC, ∠B=90°. Altitude BD to AC. If AD=4 cm, DC=9 cm, find BD.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "G is centroid of ΔABC. If AG=6 cm, find length of median from A.", "options": {"A": "9 cm", "B": "12 cm", "C": "15 cm", "D": "18 cm"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "In equilateral ΔABC, altitude = 6√3 cm. Find side length.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "D"},
    {"id": 25, "difficulty": "medium", "question": "In ΔABC, medians BE and CF intersect at G. If GF=4 cm, find GC.", "options": {"A": "4 cm", "B": "6 cm", "C": "8 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 26, "difficulty": "medium", "question": "In isosceles ΔABC, AB=AC=13 cm, BC=10 cm. Find altitude from A.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "C"},
    {"id": 27, "difficulty": "medium", "question": "Area of triangle = 48 cm², base = 12 cm. Find altitude.", "options": {"A": "4 cm", "B": "6 cm", "C": "8 cm", "D": "10 cm"}, "correct_answer": "C"},
    {"id": 28, "difficulty": "medium", "question": "In ΔPQR, altitude PS=8 cm, QR=14 cm. Find area.", "options": {"A": "56 cm²", "B": "64 cm²", "C": "72 cm²", "D": "112 cm²"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Centroid divides triangle into 6 equal areas. If area ΔABC=36 cm², area ΔGBC=?", "options": {"A": "6 cm²", "B": "9 cm²", "C": "12 cm²", "D": "18 cm²"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "In right triangle, altitude to hypotenuse divides it into segments 3 cm and 12 cm. Find altitude.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "In ΔABC, median AD=12 cm. Distance from centroid to BC=?", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "In obtuse ΔABC, ∠B>90°. Altitude from B to AC:", "options": {"A": "Lies inside", "B": "Lies outside", "C": "Is BC itself", "D": "Is AB itself"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "In ΔXYZ, altitudes from X and Y intersect at O. If ∠X=50°, ∠Y=70°, find ∠XOZ.", "options": {"A": "60°", "B": "70°", "C": "80°", "D": "90°"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "If three medians are 9 cm, 12 cm, 15 cm, can triangle exist?", "options": {"A": "Yes", "B": "No", "C": "Only if right", "D": "Only if obtuse"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "In ΔABC, AB=8 cm, BC=15 cm, CA=17 cm. Find altitude to longest side.", "options": {"A": "6 cm", "B": "7.06 cm", "C": "8 cm", "D": "9 cm"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "G is centroid of ΔABC. If area ΔABG=10 cm², area ΔABC=?", "options": {"A": "20 cm²", "B": "30 cm²", "C": "40 cm²", "D": "60 cm²"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "In equilateral triangle, altitude = median = angle bisector. If side=10 cm, find altitude.", "options": {"A": "5√3 cm", "B": "5√2 cm", "C": "10√3 cm", "D": "10√2 cm"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "In right ΔABC, altitude BD to hypotenuse AC. If AB=6 cm, BC=8 cm, find BD.", "options": {"A": "3.6 cm", "B": "4.8 cm", "C": "5.2 cm", "D": "6.4 cm"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "Medians of right triangle intersect at G. If hypotenuse=20 cm, distance from G to hypotenuse=?", "options": {"A": "5 cm", "B": "10/3 cm", "C": "20/3 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "In ΔPQR, altitude from P=10 cm, QR=16 cm. If area=80 cm², find PS.", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "In scalene triangle, which is always inside triangle?", "options": {"A": "Altitude", "B": "Median", "C": "Both", "D": "Neither"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "medium", "question": "If altitude from A=12 cm and BC=18 cm, area ΔABC=?", "options": {"A": "108 cm²", "B": "120 cm²", "C": "144 cm²", "D": "216 cm²"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "In ΔABC, AD is median. If AB=10 cm, AC=14 cm, and AD=6 cm, find BC.", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "14 cm"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "medium", "question": "Centroid coordinates: A(0,0), B(4,0), C(0,6). Find centroid.", "options": {"A": "(4/3, 2)", "B": "(2, 4/3)", "C": "(3, 4)", "D": "(4, 3)"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "In isosceles right triangle, altitude to hypotenuse=5 cm. Find hypotenuse.", "options": {"A": "5√2 cm", "B": "10 cm", "C": "10√2 cm", "D": "20 cm"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "medium", "question": "Median to hypotenuse in right triangle = 10 cm. Find hypotenuse.", "options": {"A": "10 cm", "B": "15 cm", "C": "20 cm", "D": "25 cm"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "medium", "question": "In ΔABC, altitudes from B and C are equal. Triangle is:", "options": {"A": "Isosceles", "B": "Equilateral", "C": "Right", "D": "Scalene"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "medium", "question": "G is centroid. If distance from G to BC=4 cm, find altitude from A to BC.", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "16 cm"}, "correct_answer": "C"},
    {"id": 49, "difficulty": "medium", "question": "In ΔPQR, ∠P=90°. Altitude PS to QR. If QS=4 cm, SR=9 cm, find PR.", "options": {"A": "6 cm", "B": "9 cm", "C": "12 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "Area of triangle formed by medians of 6 cm, 8 cm, 10 cm triangle?", "options": {"A": "12 cm²", "B": "16 cm²", "C": "20 cm²", "D": "24 cm²"}, "correct_answer": "B"},
    {"id": 51, "difficulty": "medium", "question": "In equilateral triangle, centroid, orthocenter, circumcenter, incenter:", "options": {"A": "All different", "B": "All same", "C": "Three same", "D": "Two same"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "medium", "question": "In ΔABC, median AD=18 cm. If G is centroid, find AG.", "options": {"A": "6 cm", "B": "9 cm", "C": "12 cm", "D": "15 cm"}, "correct_answer": "C"},
    {"id": 53, "difficulty": "medium", "question": "Altitude from right angle vertex in right triangle divides hypotenuse in ratio 1:3. If altitude=6 cm, find hypotenuse.", "options": {"A": "12 cm", "B": "16 cm", "C": "20 cm", "D": "24 cm"}, "correct_answer": "D"},
    {"id": 54, "difficulty": "medium", "question": "In ΔXYZ, XY=15 cm, YZ=20 cm, ZX=25 cm. Find altitude to longest side.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "medium", "question": "If area=54 cm² and altitude=9 cm, find base.", "options": {"A": "6 cm", "B": "9 cm", "C": "12 cm", "D": "18 cm"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "In acute triangle, orthocenter and circumcenter:", "options": {"A": "Coincide", "B": "Are inside", "C": "Orthocenter inside, circumcenter may be outside", "D": "Both inside"}, "correct_answer": "D"},
    {"id": 57, "difficulty": "medium", "question": "Median to hypotenuse in right triangle equals:", "options": {"A": "Half of hypotenuse", "B": "Equal to hypotenuse", "C": "Twice hypotenuse", "D": "Half of leg"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "In ΔABC, altitudes from A and B intersect at H. If ∠ACB=40°, find ∠AHB.", "options": {"A": "100°", "B": "120°", "C": "140°", "D": "160°"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "medium", "question": "If median length formula: m_a=½√(2b²+2c²-a²). For sides 7,8,9, median to side 9=?", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "In right triangle with legs 6 cm, 8 cm, find distance from centroid to hypotenuse.", "options": {"A": "2 cm", "B": "2.4 cm", "C": "3 cm", "D": "3.6 cm"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "medium", "question": "In ΔPQR, PS is altitude. If ∠Q=50°, ∠R=70°, find ∠QPS.", "options": {"A": "20°", "B": "30°", "C": "40°", "D": "50°"}, "correct_answer": "C"},
    {"id": 62, "difficulty": "medium", "question": "Medians of triangle are concurrent. Ratio in which they divide each other:", "options": {"A": "1:1", "B": "1:2", "C": "2:1", "D": "3:1"}, "correct_answer": "C"},
    {"id": 63, "difficulty": "medium", "question": "In isosceles triangle, altitude to base=12 cm, base=10 cm. Find equal sides.", "options": {"A": "13 cm", "B": "14 cm", "C": "15 cm", "D": "16 cm"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "If orthocenter and centroid coincide, triangle is:", "options": {"A": "Right", "B": "Isosceles", "C": "Equilateral", "D": "Scalene"}, "correct_answer": "C"},
    {"id": 65, "difficulty": "medium", "question": "In ΔABC, AD ⟂ BC. If AB=13 cm, AC=15 cm, BD=5 cm, find DC.", "options": {"A": "9 cm", "B": "10 cm", "C": "11 cm", "D": "12 cm"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "Area of triangle with medians 9,12,15 cm?", "options": {"A": "54 cm²", "B": "72 cm²", "C": "90 cm²", "D": "108 cm²"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "In obtuse triangle, orthocenter lies:", "options": {"A": "Inside", "B": "On triangle", "C": "Outside", "D": "At vertex"}, "correct_answer": "C"},
    {"id": 68, "difficulty": "medium", "question": "Centroid formula: ((x1+x2+x3)/3, (y1+y2+y3)/3). For vertices (0,0), (6,0), (0,8), centroid?", "options": {"A": "(2, 8/3)", "B": "(3, 4)", "C": "(2, 3)", "D": "(3, 8/3)"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "In right triangle, altitude to hypotenuse creates similar triangles. If segments are 2 cm and 8 cm, find altitude.", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "medium", "question": "In ΔXYZ, median XM=15 cm. If G is centroid, find XG.", "options": {"A": "5 cm", "B": "7.5 cm", "C": "10 cm", "D": "12.5 cm"}, "correct_answer": "C"},
    {"id": 71, "difficulty": "medium", "question": "In equilateral triangle of side 8 cm, find distance from centroid to vertex.", "options": {"A": "8/√3 cm", "B": "8√3/3 cm", "C": "4√3 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 72, "difficulty": "medium", "question": "If altitude from A=10 cm and area=60 cm², find BC.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "D"},
    {"id": 73, "difficulty": "medium", "question": "In ΔABC, altitudes from B and C are 8 cm and 12 cm. If BC=16 cm, find area.", "options": {"A": "64 cm²", "B": "80 cm²", "C": "96 cm²", "D": "128 cm²"}, "correct_answer": "C"},
    {"id": 74, "difficulty": "medium", "question": "Median to hypotenuse in right triangle=7.5 cm. Find hypotenuse.", "options": {"A": "10 cm", "B": "12.5 cm", "C": "15 cm", "D": "20 cm"}, "correct_answer": "C"},
    {"id": 75, "difficulty": "medium", "question": "In isosceles triangle, altitude to base=15 cm, base=24 cm. Find perimeter.", "options": {"A": "50 cm", "B": "60 cm", "C": "70 cm", "D": "80 cm"}, "correct_answer": "B"},
    {"id": 76, "difficulty": "medium", "question": "G is centroid. Area ΔABC=36 cm². Area quadrilateral formed by medians through G?", "options": {"A": "12 cm²", "B": "18 cm²", "C": "24 cm²", "D": "27 cm²"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "In right triangle, orthocenter at right angle vertex. If legs 5 cm, 12 cm, find distance from orthocenter to hypotenuse.", "options": {"A": "60/13 cm", "B": "5 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "In ΔPQR, altitude PS=12 cm. If ∠Q=45°, find PQ.", "options": {"A": "12√2 cm", "B": "12 cm", "C": "6√2 cm", "D": "24 cm"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "If medians of triangle are 6 cm, 8 cm, 10 cm, area of triangle?", "options": {"A": "24 cm²", "B": "32 cm²", "C": "40 cm²", "D": "48 cm²"}, "correct_answer": "B"},
    {"id": 80, "difficulty": "medium", "question": "In acute triangle, altitudes intersect inside. If one altitude=14 cm and divides side into 3 cm and 4 cm, find area.", "options": {"A": "42 cm²", "B": "49 cm²", "C": "56 cm²", "D": "63 cm²"}, "correct_answer": "B"},
    {"id": 81, "difficulty": "hard", "question": "In ΔABC, AD, BE, CF are medians intersecting at G. If area ΔABC=72 cm², find area ΔAFG.", "options": {"A": "6 cm²", "B": "8 cm²", "C": "10 cm²", "D": "12 cm²"}, "correct_answer": "D"},
  {"id": 82, "difficulty": "hard", "question": "In right ΔABC, ∠B=90°. Altitude BD to AC. If AD=9 cm and DC=16 cm (valid in any right triangle altitude theorem), find ratio BD:AC.", "options": {"A": "12:25", "B": "3:5", "C": "4:5", "D": "9:16"}, "correct_answer": "A"}
,  {"id": 83, "difficulty": "hard", "question": "Medians of triangle with sides 13,14,15 cm. Find length of median to side 14 cm.", "options": {"A": "√130 cm", "B": "√131 cm", "C": "√132 cm", "D": "√133 cm"}, "correct_answer": "B"},
    {"id": 84, "difficulty": "hard", "question": "In ΔABC, altitudes from B and C are 12 cm and 15 cm. If BC=20 cm and area=150 cm², find AB.", "options": {"A": "13 cm", "B": "15 cm", "C": "17 cm", "D": "20 cm"}, "correct_answer": "B"},
    {"id": 85, "difficulty": "hard", "question": "G is centroid of ΔABC. If AB²+BC²+CA²=300 cm² and AG²+BG²+CG²=100 cm², find AB².", "options": {"A": "50", "B": "75", "C": "100", "D": "125"}, "correct_answer": "C"},
    {"id": 86, "difficulty": "hard", "question": "In obtuse ΔABC, ∠A>90°. Altitudes from B and C intersect at H outside. If ∠BAC=120°, find ∠BHC.", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Medians of triangle are 6√3 cm each. Find area of triangle.", "options": {"A": "27 cm²", "B": "36 cm²", "C": "54 cm²", "D": "72 cm²"}, "correct_answer": "C"},
    {"id": 88, "difficulty": "hard", "question": "In ΔABC, AD is altitude, AE is median. If ∠BAC=60°, AB=8 cm, AC=12 cm, find DE.", "options": {"A": "2 cm", "B": "√7 cm", "C": "√10 cm", "D": "√13 cm"}, "correct_answer": "C"},
    {"id": 89, "difficulty": "hard", "question": "Orthocenter H of acute ΔABC. If ∠BAC=50°, find ∠BHC.", "options": {"A": "100°", "B": "130°", "C": "140°", "D": "150°"}, "correct_answer": "B"},
    {"id": 90, "difficulty": "hard", "question": "In ΔPQR, medians are 9 cm, 12 cm, 15 cm. Find area of triangle whose sides are medians.", "options": {"A": "36 cm²", "B": "48 cm²", "C": "54 cm²", "D": "72 cm²"}, "correct_answer": "D"},
    {"id": 91, "difficulty": "hard", "question": "In right triangle, distance from centroid to hypotenuse is 4 cm. If one leg=12 cm, find other leg.", "options": {"A": "9 cm", "B": "12 cm", "C": "15 cm", "D": "16 cm"}, "correct_answer": "D"},
    {"id": 92, "difficulty": "hard", "question": "In ΔABC, altitudes AD=12 cm, BE=15 cm, CF=20 cm. If BC=25 cm, find AB×AC.", "options": {"A": "300 cm²", "B": "400 cm²", "C": "500 cm²", "D": "600 cm²"}, "correct_answer": "C"},
    {"id": 93, "difficulty": "hard", "question": "Medians of triangle divide it into 6 equal areas. If area ΔABC=96 cm², find area of triangle formed by midpoints of medians.", "options": {"A": "12 cm²", "B": "16 cm²", "C": "18 cm²", "D": "24 cm²"}, "correct_answer": "B"},
    {"id": 94, "difficulty": "hard", "question": "In ΔXYZ, ∠X=45°, ∠Y=60°. Altitude from Z divides XY into segments with ratio:", "options": {"A": "1:√3", "B": "√3:1", "C": "1:√2", "D": "√2:1"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "If centroid divides median in ratio 2:1 from vertex. For triangle with vertices (x1,y1), (x2,y2), (x3,y3), point dividing A and midpoint of BC in ratio 2:1 is:", "options": {"A": "Centroid", "B": "Orthocenter", "C": "Circumcenter", "D": "Incenter"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "In acute ΔABC, orthocenter H. If ∠BAC=65°, ∠ABC=55°, find ∠AHC.", "options": {"A": "110°", "B": "120°", "C": "130°", "D": "140°"}, "correct_answer": "B"},
    {"id": 97, "difficulty": "hard", "question": "In ΔPQR, median PS=12 cm. If QS=5 cm, PR=13 cm, find PQ.", "options": {"A": "10 cm", "B": "12 cm", "C": "13 cm", "D": "15 cm"}, "correct_answer": "D"},
    {"id": 98, "difficulty": "hard", "question": "Altitude to hypotenuse in right triangle divides it into two triangles with areas 24 cm² and 54 cm². Find area of original triangle.", "options": {"A": "78 cm²", "B": "96 cm²", "C": "108 cm²", "D": "150 cm²"}, "correct_answer": "A"}
, {"id": 99, "difficulty": "hard", "question": "In ΔABC, G is centroid. If area ΔABG=15 cm² and area ΔACG=20 cm², find area ΔABC.", "options": {"A": "60 cm²", "B": "75 cm²", "C": "90 cm²", "D": "105 cm²"}, "correct_answer": "D"},
    {"id": 100, "difficulty": "hard", "question": "In right triangle with legs 3 cm and 4 cm, find distance from orthocenter to centroid.", "options": {"A": "5/6 cm", "B": "5/3 cm", "C": "10/3 cm", "D": "5 cm"}, "correct_answer": "B"}
  ,
{"id":101,"difficulty":"easy","question":"A median always joins a vertex to the _____ of the opposite side.","options":{"A":"Midpoint","B":"Endpoint","C":"Foot of altitude","D":"Angle bisector"},"correct_answer":"A"},
    {"id":102,"difficulty":"easy","question":"An altitude is always drawn at _____ angle to the base.","options":{"A":"30°","B":"45°","C":"60°","D":"90°"},"correct_answer":"D"},
    {"id":103,"difficulty":"easy","question":"Number of medians in any triangle is:","options":{"A":"1","B":"2","C":"3","D":"4"},"correct_answer":"C"},
    {"id":104,"difficulty":"easy","question":"The point of intersection of medians is called:","options":{"A":"Orthocenter","B":"Centroid","C":"Incenter","D":"Circumcenter"},"correct_answer":"B"},
    {"id":105,"difficulty":"easy","question":"In an obtuse triangle, orthocenter lies:","options":{"A":"Inside","B":"On the side","C":"Outside","D":"At centroid"},"correct_answer":"C"},
    {"id":106,"difficulty":"easy","question":"Median divides a triangle into two triangles of equal:","options":{"A":"Sides","B":"Angles","C":"Areas","D":"Perimeters"},"correct_answer":"C"},
    {"id":107,"difficulty":"easy","question":"Altitude is mainly used to find:","options":{"A":"Perimeter","B":"Area","C":"Angle","D":"Median"},"correct_answer":"B"},
    {"id":108,"difficulty":"easy","question":"In equilateral triangle, median and altitude are:","options":{"A":"Different","B":"Parallel","C":"Same","D":"Perpendicular"},"correct_answer":"C"},
    {"id":109,"difficulty":"easy","question":"Centroid divides each median in ratio:","options":{"A":"1:1","B":"1:2","C":"2:1","D":"3:2"},"correct_answer":"C"},
    {"id":110,"difficulty":"easy","question":"In right triangle, orthocenter lies at:","options":{"A":"Midpoint of hypotenuse","B":"Right angle vertex","C":"Centroid","D":"Outside"},"correct_answer":"B"},

    {"id":111,"difficulty":"medium","question":"In ΔABC, AD is a median. If area of ΔABC = 30 cm², area of ΔABD = ?","options":{"A":"10","B":"12","C":"15","D":"20"},"correct_answer":"C"},
    {"id":112,"difficulty":"medium","question":"If length of median is 9 cm, distance from centroid to midpoint of opposite side is:","options":{"A":"3","B":"4","C":"5","D":"6"},"correct_answer":"A"},
    {"id":113,"difficulty":"medium","question":"In a triangle, if altitude is 12 cm and base is 10 cm, area is:","options":{"A":"60","B":"120","C":"72","D":"48"},"correct_answer":"A"},
    {"id":114,"difficulty":"medium","question":"In right triangle, altitude to hypotenuse divides it into segments 4 cm and 9 cm. Length of altitude is:","options":{"A":"5","B":"6","C":"7","D":"8"},"correct_answer":"B"},
    {"id":115,"difficulty":"medium","question":"If centroid divides median in ratio 2:1, which part is longer?","options":{"A":"Vertex to centroid","B":"Centroid to base","C":"Both equal","D":"Cannot say"},"correct_answer":"A"},
    {"id":116,"difficulty":"medium","question":"Area of triangle is 40 cm² and base is 8 cm. Altitude is:","options":{"A":"4","B":"5","C":"6","D":"10"},"correct_answer":"B"},
    {"id":117,"difficulty":"medium","question":"In an isosceles triangle, altitude to base is also:","options":{"A":"Median","B":"Angle bisector","C":"Perpendicular bisector","D":"All of these"},"correct_answer":"D"},
    {"id":118,"difficulty":"medium","question":"In right triangle, median to hypotenuse is equal to:","options":{"A":"Half hypotenuse","B":"Hypotenuse","C":"One leg","D":"Twice hypotenuse"},"correct_answer":"A"},
    {"id":119,"difficulty":"medium","question":"If median from vertex is 12 cm, length of full median is:","options":{"A":"18","B":"24","C":"12","D":"6"},"correct_answer":"A"},
    {"id":120,"difficulty":"medium","question":"In scalene triangle, which is always inside the triangle?","options":{"A":"Altitude","B":"Median","C":"Orthocenter","D":"Circumcenter"},"correct_answer":"B"},

    {"id":121,"difficulty":"medium","question":"If centroid G divides median AD and AG = 8 cm, AD = ?","options":{"A":"10","B":"12","C":"14","D":"16"},"correct_answer":"B"},
    {"id":122,"difficulty":"medium","question":"Area of triangle is 54 cm² and altitude is 9 cm. Base is:","options":{"A":"6","B":"9","C":"12","D":"18"},"correct_answer":"C"},
    {"id":123,"difficulty":"medium","question":"In right triangle with legs 6 cm and 8 cm, altitude to hypotenuse is:","options":{"A":"3.6","B":"4.8","C":"5","D":"6"},"correct_answer":"B"},
    {"id":124,"difficulty":"medium","question":"If medians of triangle intersect at G, number of small triangles formed is:","options":{"A":"3","B":"4","C":"6","D":"9"},"correct_answer":"C"},
    {"id":125,"difficulty":"medium","question":"In equilateral triangle of side 12 cm, length of altitude is:","options":{"A":"6√3","B":"12√3","C":"3√3","D":"4√3"},"correct_answer":"A"},

    {"id":126,"difficulty":"hard","question":"In ΔABC, medians intersect at G. If area of ΔABC is 72 cm², area of ΔABG is:","options":{"A":"12","B":"18","C":"24","D":"36"},"correct_answer":"A"},
    {"id":127,"difficulty":"hard","question":"Altitude to hypotenuse in right triangle divides it into segments 9 cm and 16 cm. Length of hypotenuse is:","options":{"A":"20","B":"25","C":"30","D":"35"},"correct_answer":"B"},
    {"id":128,"difficulty":"hard","question":"If medians of triangle are equal, triangle must be:","options":{"A":"Right","B":"Isosceles","C":"Equilateral","D":"Scalene"},"correct_answer":"C"},
    {"id":129,"difficulty":"hard","question":"In acute triangle, orthocenter lies inside. Which angle is always less than 90°?","options":{"A":"Angle at centroid","B":"Angle at orthocenter","C":"All angles","D":"Only base angles"},"correct_answer":"C"},
    {"id":130,"difficulty":"hard","question":"Area of triangle formed by joining midpoints of sides is what fraction of original?","options":{"A":"1/2","B":"1/3","C":"1/4","D":"1/6"},"correct_answer":"C"},
    {"id":131,"difficulty":"hard","question":"If centroid and orthocenter coincide, triangle is:","options":{"A":"Right","B":"Isosceles","C":"Equilateral","D":"Scalene"},"correct_answer":"C"},
    {"id":132,"difficulty":"hard","question":"In right triangle, distance from centroid to hypotenuse is 5 cm. Hypotenuse length is:","options":{"A":"10","B":"15","C":"20","D":"30"},"correct_answer":"C"},
    {"id":133,"difficulty":"hard","question":"Medians of triangle divide it into 6 equal triangles. Ratio of areas is:","options":{"A":"1:1","B":"2:1","C":"3:1","D":"6:1"},"correct_answer":"A"},
    {"id":134,"difficulty":"hard","question":"If altitude = 10 cm and area = 80 cm², base = ?","options":{"A":"8","B":"10","C":"12","D":"16"},"correct_answer":"D"},
    {"id":135,"difficulty":"hard","question":"In ΔABC, altitude from A divides BC in ratio 2:3. If BC = 15 cm, the segments are:","options":{"A":"6,9","B":"5,10","C":"4,11","D":"3,12"},"correct_answer":"A"},
    {"id":136,"difficulty":"hard","question":"Which center is point of concurrency of altitudes?","options":{"A":"Centroid","B":"Orthocenter","C":"Incenter","D":"Circumcenter"},"correct_answer":"B"},
    {"id":137,"difficulty":"hard","question":"In obtuse triangle, how many altitudes lie outside the triangle?","options":{"A":"0","B":"1","C":"2","D":"3"},"correct_answer":"C"},
    {"id":138,"difficulty":"hard","question":"If triangle has area 96 cm², area of triangle formed by medians is:","options":{"A":"16","B":"24","C":"32","D":"48"},"correct_answer":"B"},
    {"id":139,"difficulty":"hard","question":"Distance between centroid and vertex is 10 cm. Length of median is:","options":{"A":"12","B":"15","C":"18","D":"20"},"correct_answer":"B"},
    {"id":140,"difficulty":"hard","question":"In right triangle with sides 5 cm, 12 cm, 13 cm, distance from centroid to hypotenuse is:","options":{"A":"13/6","B":"5/2","C":"10/3","D":"13/3"},"correct_answer":"A"},
    {"id":141,"difficulty":"hard","question":"If area of ΔABC = 90 cm², area of triangle formed by joining midpoints of medians is:","options":{"A":"10","B":"15","C":"20","D":"30"},"correct_answer":"B"},
    {"id":142,"difficulty":"hard","question":"Which of the following always lies inside any triangle?","options":{"A":"Altitude","B":"Median","C":"Orthocenter","D":"Circumcenter"},"correct_answer":"B"},
    {"id":143,"difficulty":"hard","question":"If altitudes from two vertices are equal, triangle is:","options":{"A":"Scalene","B":"Right","C":"Isosceles","D":"Equilateral"},"correct_answer":"C"},
    {"id":144,"difficulty":"hard","question":"In equilateral triangle of side 6 cm, distance from centroid to base is:","options":{"A":"√3","B":"2√3","C":"3√3","D":"4√3"},"correct_answer":"A"},
    {"id":145,"difficulty":"hard","question":"If area = 48 cm² and median to base is 8 cm, base length is:","options":{"A":"6","B":"8","C":"12","D":"16"},"correct_answer":"C"},
    {"id":146,"difficulty":"hard","question":"In triangle, altitude and median coincide only in:","options":{"A":"Right triangle","B":"Isosceles triangle","C":"Equilateral triangle","D":"Scalene triangle"},"correct_answer":"B"},
    {"id":147,"difficulty":"hard","question":"If centroid divides median in ratio 2:1, then ratio of distances from centroid to vertex and base is:","options":{"A":"1:2","B":"2:1","C":"3:1","D":"1:3"},"correct_answer":"B"},
    {"id":148,"difficulty":"hard","question":"In triangle, if altitude is longest, triangle is likely:","options":{"A":"Acute","B":"Right","C":"Obtuse","D":"Equilateral"},"correct_answer":"A"},
    {"id":149,"difficulty":"hard","question":"Area of triangle formed by joining midpoints of sides of a triangle with area 64 cm² is:","options":{"A":"8","B":"16","C":"24","D":"32"},"correct_answer":"B"},
    {"id":150,"difficulty":"hard","question":"If centroid, circumcenter and orthocenter coincide, triangle must be:","options":{"A":"Right","B":"Isosceles","C":"Scalene","D":"Equilateral"},"correct_answer":"D"}
  ]
}