{
  "title": "Properties of Isosceles Triangle - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "In an isosceles triangle, if the vertex angle is 80°, what is each base angle?", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "80°"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "Which of these is always true for an isosceles triangle?", "options": {"A": "All sides equal", "B": "Two sides equal", "C": "All angles equal", "D": "No sides equal"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "In isosceles triangle ABC with AB = AC, which angles are equal?", "options": {"A": "∠A = ∠B", "B": "∠B = ∠C", "C": "∠A = ∠C", "D": "All equal"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "If base angles of isosceles triangle are 65° each, what is vertex angle?", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "In isosceles triangle, altitude to base also acts as ______.", "options": {"A": "median", "B": "angle bisector", "C": "perpendicular bisector", "D": "All of these"}, "correct_answer": "D"},
    {"id": 6, "difficulty": "easy", "question": "If two sides of triangle are 7 cm and 7 cm, what type of triangle?", "options": {"A": "Scalene", "B": "Isosceles", "C": "Equilateral", "D": "Right"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "In isosceles ΔPQR with PQ = PR, if ∠Q = 70°, find ∠R.", "options": {"A": "60°", "B": "70°", "C": "80°", "D": "90°"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "An equilateral triangle is also ______ triangle.", "options": {"A": "isosceles", "B": "right", "C": "obtuse", "D": "scalene"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "In isosceles triangle, equal sides are 10 cm each, base is 8 cm. Find perimeter.", "options": {"A": "26 cm", "B": "28 cm", "C": "30 cm", "D": "32 cm"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "If vertex angle of isosceles triangle is 100°, each base angle is ______.", "options": {"A": "30°", "B": "40°", "C": "50°", "D": "60°"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "In ΔABC, AB = AC = 6 cm, BC = 4 cm. Which is base?", "options": {"A": "AB", "B": "BC", "C": "AC", "D": "None"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "Which triangle has two equal angles?", "options": {"A": "Scalene", "B": "Isosceles", "C": "Right", "D": "Obtuse"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "In isosceles triangle, median to base is also ______.", "options": {"A": "altitude", "B": "perpendicular", "C": "hypotenuse", "D": "side"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "If isosceles triangle has perimeter 32 cm and equal sides 10 cm each, find base.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "In isosceles right triangle, what are the angles?", "options": {"A": "45°, 45°, 90°", "B": "30°, 60°, 90°", "C": "60°, 60°, 60°", "D": "50°, 50°, 80°"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Which statement is false for isosceles triangle?", "options": {"A": "Two sides equal", "B": "Two angles equal", "C": "Altitude to base bisects base", "D": "All medians equal"}, "correct_answer": "D"},
    {"id": 17, "difficulty": "easy", "question": "If base of isosceles triangle is 12 cm and each equal side 10 cm, find height to base.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "In ΔXYZ, XY = XZ = 8 cm, ∠Y = 55°. Find ∠Z.", "options": {"A": "55°", "B": "65°", "C": "70°", "D": "75°"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Isosceles triangle can be ______ triangle.", "options": {"A": "acute", "B": "right", "C": "obtuse", "D": "All of these"}, "correct_answer": "D"},
    {"id": 20, "difficulty": "easy", "question": "If equal sides of isosceles triangle are 13 cm each, base 10 cm, find area using formula ½ × base × height.", "options": {"A": "50 cm²", "B": "60 cm²", "C": "65 cm²", "D": "70 cm²"}, "correct_answer": "B"},
    
    {"id": 21, "difficulty": "medium", "question": "In isosceles triangle ABC with AB=AC, AD⊥BC. If BD=4 cm and AD=3 cm, find AB.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Vertex angle of isosceles triangle is 40°. Find each base angle.", "options": {"A": "60°", "B": "65°", "C": "70°", "D": "75°"}, "correct_answer": "C"},
    {"id": 23, "difficulty": "medium", "question": "In ΔPQR, PQ=PR. If ∠P=80°, find ∠Q.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "80°"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "Isosceles triangle has sides 8 cm, 8 cm, 5 cm. Find altitude to base of 5 cm.", "options": {"A": "√55 cm", "B": "√57 cm", "C": "√59 cm", "D": "√61 cm"}, "correct_answer": "C"},
    {"id": 25, "difficulty": "medium", "question": "In isosceles ΔABC, AB=AC=10 cm, BC=12 cm. Find length of median from A to BC.", "options": {"A": "6 cm", "B": "7 cm", "C": "8 cm", "D": "9 cm"}, "correct_answer": "C"},
    {"id": 26, "difficulty": "medium", "question": "If two angles of triangle are 55° and 55°, what type of triangle?", "options": {"A": "Scalene", "B": "Isosceles", "C": "Equilateral", "D": "Right"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "In ΔXYZ, XY=XZ. If altitude from X to YZ is 6 cm and YZ=8 cm, find XY.", "options": {"A": "7 cm", "B": "8 cm", "C": "9 cm", "D": "10 cm"}, "correct_answer": "D"},
    {"id": 28, "difficulty": "medium", "question": "Base of isosceles triangle is 16 cm. Each equal side is 10 cm. Find area.", "options": {"A": "36 cm²", "B": "42 cm²", "C": "48 cm²", "D": "54 cm²"}, "correct_answer": "C"},
    {"id": 29, "difficulty": "medium", "question": "In isosceles triangle, if base angle is twice vertex angle, find all angles.", "options": {"A": "36°, 72°, 72°", "B": "40°, 70°, 70°", "C": "45°, 67.5°, 67.5°", "D": "30°, 75°, 75°"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "ΔABC is isosceles with AB=AC. If ∠B=2x+10° and ∠C=3x-5°, find ∠A.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "D"},
    {"id": 31, "difficulty": "medium", "question": "In isosceles right triangle, hypotenuse is 10√2 cm. Find each leg.", "options": {"A": "8 cm", "B": "9 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 32, "difficulty": "medium", "question": "Perimeter of isosceles triangle is 32 cm. Base is 12 cm. Find equal sides.", "options": {"A": "8 cm each", "B": "9 cm each", "C": "10 cm each", "D": "11 cm each"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "In ΔPQR, PQ=PR. If altitude PS=12 cm and QR=10 cm, find PQ.", "options": {"A": "13 cm", "B": "14 cm", "C": "15 cm", "D": "16 cm"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Isosceles triangle has vertex angle 120°. If each equal side is 8 cm, find base.", "options": {"A": "8√2 cm", "B": "8√3 cm", "C": "12 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "In ΔABC, AB=AC. If AD is median to BC and AD=6 cm, BC=8 cm, find AB.", "options": {"A": "√52 cm", "B": "√53 cm", "C": "√54 cm", "D": "√55 cm"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "If vertex angle of isosceles triangle is 36°, find each base angle.", "options": {"A": "72°", "B": "74°", "C": "76°", "D": "78°"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "In isosceles triangle, altitude to base divides triangle into two ______ triangles.", "options": {"A": "congruent right", "B": "similar", "C": "equilateral", "D": "isosceles"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "ΔXYZ is isosceles with XY=XZ=13 cm, YZ=10 cm. Find distance from X to YZ.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "C"},
    {"id": 39, "difficulty": "medium", "question": "If base angles of isosceles triangle are (3x+5)° and (4x-15)°, find vertex angle.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "D"},
    {"id": 40, "difficulty": "medium", "question": "In isosceles triangle, equal sides are 17 cm each, base 16 cm. Find area using Heron's formula.", "options": {"A": "120 cm²", "B": "125 cm²", "C": "130 cm²", "D": "135 cm²"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "Triangle has angles 80°, 50°, 50°. What type of triangle?", "options": {"A": "Scalene", "B": "Isosceles", "C": "Equilateral", "D": "Right"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "medium", "question": "In ΔABC, AB=AC, ∠A=100°. Which side is shortest?", "options": {"A": "AB", "B": "BC", "C": "AC", "D": "All equal"}, "correct_answer": "B"},
    {"id": 43, "difficulty": "medium", "question": "Isosceles triangle perimeter is 50 cm. If base is 14 cm, find equal sides.", "options": {"A": "16 cm each", "B": "17 cm each", "C": "18 cm each", "D": "19 cm each"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "medium", "question": "In right isosceles triangle, if leg is 7 cm, find hypotenuse.", "options": {"A": "7√2 cm", "B": "7√3 cm", "C": "14 cm", "D": "10 cm"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "ΔPQR is isosceles with PQ=PR. If ∠Q=65°, find ∠P.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "medium", "question": "In isosceles triangle, if vertex angle is 90°, what are base angles?", "options": {"A": "30° each", "B": "45° each", "C": "60° each", "D": "75° each"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "medium", "question": "Equal sides of isosceles triangle are 25 cm each, base 14 cm. Find altitude.", "options": {"A": "20 cm", "B": "22 cm", "C": "24 cm", "D": "26 cm"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "medium", "question": "In ΔABC, AB=AC. If exterior angle at B is 140°, find ∠A.", "options": {"A": "80°", "B": "90°", "C": "100°", "D": "110°"}, "correct_answer": "C"},
    {"id": 49, "difficulty": "medium", "question": "Isosceles triangle has base 24 cm and area 120 cm². Find equal side.", "options": {"A": "12 cm", "B": "13 cm", "C": "14 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "In ΔXYZ, XY=XZ. If altitude from X makes 30° with XY, find angles of triangle.", "options": {"A": "30°, 75°, 75°", "B": "40°, 70°, 70°", "C": "50°, 65°, 65°", "D": "60°, 60°, 60°"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "medium", "question": "If isosceles triangle has sides 5x, 5x, 6x and perimeter 48 cm, find area.", "options": {"A": "48 cm²", "B": "60 cm²", "C": "72 cm²", "D": "96 cm²"}, "correct_answer": "D"},
    {"id": 52, "difficulty": "medium", "question": "In ΔABC, AB=AC. If BD is altitude and BD=8 cm, BC=12 cm, find AB.", "options": {"A": "9 cm", "B": "10 cm", "C": "11 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "medium", "question": "Vertex angle of isosceles triangle is 108°. Find each base angle.", "options": {"A": "32°", "B": "34°", "C": "36°", "D": "38°"}, "correct_answer": "C"},
    {"id": 54, "difficulty": "medium", "question": "In isosceles triangle, median to base is 8 cm, base is 12 cm. Find equal side.", "options": {"A": "9 cm", "B": "10 cm", "C": "11 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "medium", "question": "ΔPQR has PQ=PR. If ∠P=2∠Q, find angles.", "options": {"A": "45°, 67.5°, 67.5°", "B": "60°, 60°, 60°", "C": "72°, 54°, 54°", "D": "90°, 45°, 45°"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "Isosceles triangle area is 60 cm², base 10 cm. Find height.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "C"},
    {"id": 57, "difficulty": "medium", "question": "In ΔABC, AB=AC=15 cm. If BC=18 cm, find distance from A to BC.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "C"},
    {"id": 58, "difficulty": "medium", "question": "If base angle of isosceles triangle is 40°, find vertex angle.", "options": {"A": "80°", "B": "90°", "C": "100°", "D": "110°"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "medium", "question": "In isosceles triangle, altitude to base divides vertex angle into 30° and 30°. Find base angles.", "options": {"A": "60° each", "B": "65° each", "C": "70° each", "D": "75° each"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "medium", "question": "ΔXYZ is isosceles with XY=XZ=10 cm. If ∠X=60°, find area.", "options": {"A": "25√3 cm²", "B": "30√3 cm²", "C": "35√3 cm²", "D": "40√3 cm²"}, "correct_answer": "A"},
    {"id": 61, "difficulty": "medium", "question": "Isosceles triangle has perimeter 42 cm. Ratio of equal side to base is 3:2. Find base.", "options": {"A": "10.5 cm", "B": "12 cm", "C": "13.5 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "medium", "question": "In ΔABC, AB=AC. If altitude AD=6 cm and AB=10 cm, find BC.", "options": {"A": "12 cm", "B": "14 cm", "C": "16 cm", "D": "18 cm"}, "correct_answer": "C"},
    {"id": 63, "difficulty": "medium", "question": "If vertex angle is 70° in isosceles triangle, find each base angle.", "options": {"A": "50°", "B": "55°", "C": "60°", "D": "65°"}, "correct_answer": "B"},
    {"id": 64, "difficulty": "medium", "question": "Isosceles triangle equal sides are 13 cm, base 10 cm. Find inradius.", "options": {"A": "3 cm", "B": "3.5 cm", "C": "4 cm", "D": "4.5 cm"}, "correct_answer": "C"},
    {"id": 65, "difficulty": "medium", "question": "In ΔPQR, PQ=PR. If ∠Q=55°, find exterior angle at R.", "options": {"A": "115°", "B": "120°", "C": "125°", "D": "130°"}, "correct_answer": "C"},
    {"id": 66, "difficulty": "medium", "question": "Isosceles triangle area is 84 cm², base 14 cm. Find perimeter.", "options": {"A": "40 cm", "B": "42 cm", "C": "44 cm", "D": "46 cm"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "In ΔABC, AB=AC, ∠B=65°. Find ∠A.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 68, "difficulty": "medium", "question": "Right isosceles triangle has hypotenuse 10√2 cm. Find area.", "options": {"A": "25 cm²", "B": "50 cm²", "C": "75 cm²", "D": "100 cm²"}, "correct_answer": "B"},
    {"id": 69, "difficulty": "medium", "question": "Isosceles triangle base is 16 cm, equal sides 17 cm each. Find median to base.", "options": {"A": "13 cm", "B": "14 cm", "C": "15 cm", "D": "16 cm"}, "correct_answer": "C"},
    {"id": 70, "difficulty": "medium", "question": "In ΔXYZ, XY=XZ. If altitude from X divides YZ into 3 cm and 3 cm, and XY=5 cm, find ∠X.", "options": {"A": "60°", "B": "75°", "C": "90°", "D": "120°"}, "correct_answer": "C"},
    {"id": 71, "difficulty": "medium", "question": "If base angles of isosceles triangle are (2x+20)° and (3x+10)°, find x.", "options": {"A": "10", "B": "15", "C": "20", "D": "25"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Isosceles triangle has sides 10 cm, 10 cm, 12 cm. Find circumradius.", "options": {"A": "6.25 cm", "B": "6.5 cm", "C": "6.75 cm", "D": "7.0 cm"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "In ΔABC, AB=AC. If ∠A=80°, find ∠B.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 74, "difficulty": "medium", "question": "Isosceles triangle perimeter 36 cm. Base is 4 cm less than equal side. Find base.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "In right isosceles triangle, if area is 32 cm², find hypotenuse.", "options": {"A": "8 cm", "B": "8√2 cm", "C": "10 cm", "D": "10√2 cm"}, "correct_answer": "B"},
    {"id": 76, "difficulty": "medium", "question": "ΔPQR has PQ=PR. If altitude PS makes 45° with PQ, find angles of triangle.", "options": {"A": "45°, 67.5°, 67.5°", "B": "60°, 60°, 60°", "C": "75°, 52.5°, 52.5°", "D": "90°, 45°, 45°"}, "correct_answer": "D"},
    {"id": 77, "difficulty": "medium", "question": "Isosceles triangle has equal sides 15 cm, height 12 cm. Find base.", "options": {"A": "16 cm", "B": "17 cm", "C": "18 cm", "D": "19 cm"}, "correct_answer": "C"},
    {"id": 78, "difficulty": "medium", "question": "In ΔABC, AB=AC. If ∠B=70°, find exterior angle at A.", "options": {"A": "110°", "B": "120°", "C": "130°", "D": "140°"}, "correct_answer": "D"},
    {"id": 79, "difficulty": "medium", "question": "Vertex angle of isosceles triangle is 20° more than base angle. Find angles.", "options": {"A": "40°, 70°, 70°", "B": "50°, 65°, 65°", "C": "55°, 62.5°, 62.5°", "D": "60°, 60°, 60°"}, "correct_answer": "B"},
    {"id": 80, "difficulty": "medium", "question": "Isosceles triangle area 48 cm², height 6 cm. Find equal side if base is 16 cm.", "options": {"A": "8 cm", "B": "9 cm", "C": "10 cm", "D": "11 cm"}, "correct_answer": "C"},
    
    {"id": 81, "difficulty": "hard", "question": "In isosceles triangle ABC with AB=AC=13 cm, BC=10 cm. Find distance between centroid and orthocenter.", "options": {"A": "5/3 cm", "B": "7/3 cm", "C": "8/3 cm", "D": "10/3 cm"}, "correct_answer": "C"},
    {"id": 82, "difficulty": "hard", "question": "In ΔABC, AB=AC. If altitude AD=12 cm and AB=15 cm, find circumradius.", "options": {"A": "75/8 cm", "B": "75/9 cm", "C": "75/10 cm", "D": "75/11 cm"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Isosceles triangle has sides 10 cm, 10 cm, 12 cm. Find angle bisector length to base.", "options": {"A": "8 cm", "B": "9 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "In ΔPQR, PQ=PR=25 cm, QR=14 cm. Find distance from circumcenter to side QR.", "options": {"A": "6 cm", "B": "7 cm", "C": "8 cm", "D": "9 cm"}, "correct_answer": "C"},
    {"id": 85, "difficulty": "hard", "question": "Vertex angle of isosceles triangle is 36°. Find ratio of base to equal side (golden ratio problem).", "options": {"A": "(√5-1)/2", "B": "(√5+1)/2", "C": "√2/2", "D": "√3/2"}, "correct_answer": "A"},
    {"id": 86, "difficulty": "hard", "question": "In isosceles triangle ABC with AB=AC, if median BD=6 cm and AD=4 cm, find BC.", "options": {"A": "4√5 cm", "B": "6√2 cm", "C": "8 cm", "D": "10 cm"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "ΔXYZ is isosceles with XY=XZ. If inradius=3 cm and base=8 cm, find XY.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "In isosceles triangle, if base is 24 cm and altitude is 5 cm, find equal sides.", "options": {"A": "11 cm", "B": "12 cm", "C": "13 cm", "D": "14 cm"}, "correct_answer": "C"},
    {"id": 89, "difficulty": "hard", "question": "ΔABC has AB=AC. If ∠A=120° and BC=6√3 cm, find AB.", "options": {"A": "6 cm", "B": "8 cm", "C": "9 cm", "D": "10 cm"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "In isosceles triangle, medians to equal sides are 15 cm each, base 24 cm. Find area.", "options": {"A": "120 cm²", "B": "144 cm²", "C": "168 cm²", "D": "192 cm²"}, "correct_answer": "B"},
    {"id": 91, "difficulty": "hard", "question": "If in ΔABC, AB=AC and angle bisector of ∠A divides BC into 3 cm and 4 cm, find AB.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 92, "difficulty": "hard", "question": "Isosceles triangle has perimeter 2s. If base is b, show area = (b/4)√(4s²-b²). For s=15, b=14, find area.", "options": {"A": "42 cm²", "B": "56 cm²", "C": "70 cm²", "D": "84 cm²"}, "correct_answer": "D"},
    {"id": 93, "difficulty": "hard", "question": "In ΔPQR, PQ=PR. If altitude PS=8 cm and circumradius=6.25 cm, find QR.", "options": {"A": "9.6 cm", "B": "10.2 cm", "C": "11.4 cm", "D": "12.0 cm"}, "correct_answer": "D"},
    {"id": 94, "difficulty": "hard", "question": "Isosceles triangle has base 10 cm and area 60 cm². Find angle between equal sides.", "options": {"A": "sin⁻¹(0.6)", "B": "sin⁻¹(0.8)", "C": "sin⁻¹(0.9)", "D": "sin⁻¹(0.95)"}, "correct_answer": "B"},
    {"id": 95, "difficulty": "hard", "question": "In ΔABC, AB=AC=10 cm. If median from B to AC is 9 cm, find BC.", "options": {"A": "4√5 cm", "B": "6√2 cm", "C": "2√21 cm", "D": "3√10 cm"}, "correct_answer": "C"},
    {"id": 96, "difficulty": "hard", "question": "Vertex angle of isosceles triangle is 108°. If equal side is 10 cm, find length of angle bisector to base.", "options": {"A": "5(√5-1) cm", "B": "5(√5+1) cm", "C": "10(√5-1)/2 cm", "D": "10(√5+1)/2 cm"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "In isosceles triangle, distance from vertex to centroid is 8 cm. Find distance from vertex to orthocenter.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 98, "difficulty": "hard", "question": "ΔXYZ has XY=XZ=13 cm. If median XM=5 cm to YZ, find area.", "options": {"A": "50 cm²", "B": "55 cm²", "C": "60 cm²", "D": "65 cm²"}, "correct_answer": "C"},
    {"id": 99, "difficulty": "hard", "question": "In isosceles triangle, if base is 16 cm and radius of circumcircle is 10 cm, find equal sides.", "options": {"A": "2√29 cm", "B": "2√31 cm", "C": "2√33 cm", "D": "2√35 cm"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "ΔABC is isosceles with AB=AC. If ∠A=40° and inradius=3 cm, find area.", "options": {"A": "9(√5+1) cm²", "B": "9(√5-1) cm²", "C": "18(√5+1) cm²", "D": "18(√5-1) cm²"}, "correct_answer": "D"}
  ]
}