{
  "title": "Division of Line Segment - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "To divide line segment in given ratio, we use:", "options": {"A": "Perpendicular bisector", "B": "Angle bisector", "C": "Parallel lines", "D": "Arcs"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "Basic proportionality theorem is also called:", "options": {"A": "Pythagoras theorem", "B": "Thales theorem", "C": "Apollonius theorem", "D": "Euler theorem"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "If point P divides AB in ratio 3:2, then AP:PB =", "options": {"A": "2:3", "B": "3:2", "C": "3:5", "D": "5:3"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "To divide AB in ratio m:n, we draw how many rays from A?", "options": {"A": "1", "B": "2", "C": "m+n", "D": "Any number"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "On ray from A, we mark equal segments of:", "options": {"A": "Length m", "B": "Length n", "C": "Length m+n", "D": "Any convenient length"}, "correct_answer": "D"},
    {"id": 6, "difficulty": "easy", "question": "After marking points on ray, we join last point to:", "options": {"A": "A", "B": "B", "C": "Midpoint", "D": "Any point"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "Then we draw lines parallel to this joining line through:", "options": {"A": "All marked points", "B": "mth point", "C": "nth point", "D": "(m+n)th point"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "Point dividing segment in ratio 1:1 is:", "options": {"A": "End point", "B": "Midpoint", "C": "Trisection point", "D": "Centroid"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "To divide AB in ratio 2:3, total equal parts on ray:", "options": {"A": "2", "B": "3", "C": "5", "D": "6"}, "correct_answer": "C"},
    {"id": 10, "difficulty": "easy", "question": "If P divides AB externally in ratio 3:2, then AP:PB =", "options": {"A": "3:2", "B": "2:3", "C": "3:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Section formula for internal division in ratio m:n: x =", "options": {"A": "(mx₁+nx₂)/(m+n)", "B": "(mx₂+nx₁)/(m+n)", "C": "(mx₁-nx₂)/(m-n)", "D": "(mx₂-nx₁)/(m-n)"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "For external division, denominator in section formula:", "options": {"A": "m+n", "B": "m-n", "C": "n-m", "D": "mn"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "Midpoint formula: x =", "options": {"A": "(x₁+x₂)/3", "B": "(x₁+x₂)/2", "C": "(x₁-x₂)/2", "D": "2(x₁+x₂)"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "If P divides AB in ratio k:1, then k =", "options": {"A": "AP/PB", "B": "PB/AP", "C": "AB/AP", "D": "AB/PB"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "To trisect segment, we divide in ratios:", "options": {"A": "1:1:1", "B": "1:2 and 2:1", "C": "1:3 and 2:3", "D": "1:2 and 1:1"}, "correct_answer": "B"},
    {"id": 16, "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": 17, "difficulty": "easy", "question": "If A(2,3), B(8,11), midpoint coordinates:", "options": {"A": "(4,5)", "B": "(5,7)", "C": "(6,9)", "D": "(7,10)"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "Point dividing (1,2) and (4,5) in ratio 1:2 internally:", "options": {"A": "(2,3)", "B": "(3,4)", "C": "(4,5)", "D": "(5,6)"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "For construction, we use property of:", "options": {"A": "Similar triangles", "B": "Congruent triangles", "C": "Equal triangles", "D": "Any triangles"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "If P divides AB in ratio 3:7, AP/AB =", "options": {"A": "3/7", "B": "7/3", "C": "3/10", "D": "7/10"}, "correct_answer": "C"},

    {"id": 21, "difficulty": "medium", "question": "Divide line segment 8 cm in ratio 3:5 internally. Length of smaller part:", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "Construct division of 10 cm segment in ratio 2:3. Distance from one end to division point:", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "A(1,4), B(7,10). Point dividing AB in ratio 2:1 internally:", "options": {"A": "(3,6)", "B": "(4,7)", "C": "(5,8)", "D": "(6,9)"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "Divide 12 cm segment in ratio 1:2:3. Length of middle part:", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "C"},
    {"id": 25, "difficulty": "medium", "question": "P divides AB of length 15 cm such that AP:PB = 3:2. Find AP.", "options": {"A": "6 cm", "B": "7.5 cm", "C": "9 cm", "D": "10.5 cm"}, "correct_answer": "C"},
    {"id": 26, "difficulty": "medium", "question": "Construct division of 9 cm segment in ratio 4:5. Distance from division point to farther end:", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "A(-2,3), B(4,7). Point dividing externally in ratio 3:2:", "options": {"A": "(10,11)", "B": "(12,13)", "C": "(14,15)", "D": "(16,17)"}, "correct_answer": "C"},
    {"id": 28, "difficulty": "medium", "question": "Divide 7.5 cm segment in ratio 3:7. Length of larger part:", "options": {"A": "2.25 cm", "B": "3.75 cm", "C": "4.5 cm", "D": "5.25 cm"}, "correct_answer": "D"},
    {"id": 29, "difficulty": "medium", "question": "P(3,4), Q(9,10). R divides PQ in ratio 1:2. Find R.", "options": {"A": "(4,5)", "B": "(5,6)", "C": "(6,7)", "D": "(7,8)"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "Construct division of 11 cm segment in ratio 5:6. Difference between two parts:", "options": {"A": "0.5 cm", "B": "1 cm", "C": "1.5 cm", "D": "2 cm"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "A(0,0), B(12,0). Points dividing in ratio 1:2 and 2:1. Distance between these points:", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "Divide 8.4 cm segment in ratio 2:3:4. Length of smallest part:", "options": {"A": "1.4 cm", "B": "1.6 cm", "C": "1.8 cm", "D": "2.0 cm"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "P divides AB where A(2,3), B(8,9) in ratio k:1. If P is (5,6), find k.", "options": {"A": "0.5", "B": "1", "C": "1.5", "D": "2"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "Construct division of 13 cm segment in ratio 3:8. Distance from division point to midpoint:", "options": {"A": "0.5 cm", "B": "1 cm", "C": "1.5 cm", "D": "2 cm"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "A(-1,2), B(5,8). Point dividing in ratio 2:3 internally:", "options": {"A": "(1.4,4.4)", "B": "(1.8,4.8)", "C": "(2.2,5.2)", "D": "(2.6,5.6)"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Divide 10.5 cm segment in ratio 4:3. Ratio of larger to smaller:", "options": {"A": "3:4", "B": "4:3", "C": "7:3", "D": "7:4"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "P(1,1), Q(7,13). R divides PQ such that PR = 3RQ. Find R.", "options": {"A": "(2.5,4)", "B": "(3,5.5)", "C": "(4,7)", "D": "(5.5,10)"}, "correct_answer": "D"},
    {"id": 38, "difficulty": "medium", "question": "Construct division of 14 cm segment in ratio 5:9. Length from end to point that divides segment and its extension in same ratio externally:", "options": {"A": "17.5 cm", "B": "21 cm", "C": "24.5 cm", "D": "28 cm"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "A(3,7), B(9,19). Point dividing in ratio 1:3:", "options": {"A": "(4.5,10)", "B": "(5,11.5)", "C": "(5.5,13)", "D": "(6,14.5)"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Divide 16.8 cm segment in ratio 3:4:5. Sum of two smaller parts:", "options": {"A": "8.4 cm", "B": "9.8 cm", "C": "11.2 cm", "D": "12.6 cm"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "medium", "question": "P divides AB where A(0,6), B(8,0) in ratio 3:5. Find P.", "options": {"A": "(2,4.5)", "B": "(3,3.75)", "C": "(4,3)", "D": "(5,2.25)"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "medium", "question": "Construct division of 18 cm segment in ratio 7:11. Point dividing the larger part in ratio 3:4 is how far from nearest end?", "options": {"A": "7 cm", "B": "8 cm", "C": "9 cm", "D": "10 cm"}, "correct_answer": "C"},
    {"id": 43, "difficulty": "medium", "question": "A(-3,-2), B(5,6). Midpoint of segment joining points dividing AB in ratios 1:3 and 3:1:", "options": {"A": "(0,0)", "B": "(1,2)", "C": "(2,4)", "D": "(3,6)"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "Divide 22.5 cm segment in ratio 2:3:4. Middle part is what fraction of whole?", "options": {"A": "2/9", "B": "1/3", "C": "4/9", "D": "5/9"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "medium", "question": "P(4,5), Q(10,11). R divides PQ externally in ratio 4:1. Find R.", "options": {"A": "(12,13)", "B": "(14,15)", "C": "(16,17)", "D": "(18,19)"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "Construct division of 15 cm segment in ratio 4:5. Then divide each part in ratio 1:1. Distance between these new division points:", "options": {"A": "0.5 cm", "B": "0.75 cm", "C": "1 cm", "D": "1.25 cm"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "medium", "question": "A(2,4), B(8,16). Point dividing in ratio k:1 is (5,10). Find k.", "options": {"A": "0.5", "B": "1", "C": "1.5", "D": "2"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "medium", "question": "Divide 19.6 cm segment in ratio 5:9. Difference between parts:", "options": {"A": "2.8 cm", "B": "3.2 cm", "C": "3.6 cm", "D": "4.0 cm"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "medium", "question": "P(-2,1), Q(4,7). Points dividing PQ in three equal parts:", "options": {"A": "(0,3) and (2,5)", "B": "(1,4) and (3,6)", "C": "(-1,2) and (1,4)", "D": "(-0.5,2.5) and (1.5,4.5)"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "Construct division of 21 cm segment in ratio 5:8. Then divide whole segment in ratio 8:5. Distance between these division points:", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "D"},
    {"id": 51, "difficulty": "medium", "question": "A(1,1), B(4,13). Point dividing in ratio 1:2:", "options": {"A": "(2,5)", "B": "(2.5,7)", "C": "(3,9)", "D": "(3.5,11)"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "Divide 25.2 cm segment in ratio 3:4:5. Length of largest part minus smallest:", "options": {"A": "4.2 cm", "B": "5.6 cm", "C": "7.0 cm", "D": "8.4 cm"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "P(0,0), Q(12,18). R divides PQ in ratio 2:3. Find distance from R to origin.", "options": {"A": "6", "B": "6√2", "C": "6√3", "D": "6√5"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "medium", "question": "Construct division of 17 cm segment in ratio 6:7. Point dividing the line joining division point to an end in ratio 2:1 is how far from that end?", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "medium", "question": "A(-5,7), B(7,-5). Point dividing in ratio 3:5:", "options": {"A": "(-0.5,2.5)", "B": "(0.5,1.5)", "C": "(1.5,0.5)", "D": "(2.5,-0.5)"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "medium", "question": "Divide 28 cm segment in ratio 5:9. Ratio of smaller part to whole:", "options": {"A": "5:9", "B": "5:14", "C": "9:14", "D": "14:5"}, "correct_answer": "B"},
    {"id": 57, "difficulty": "medium", "question": "P(2,3), Q(8,15). R divides PQ externally in ratio 3:2. Find R.", "options": {"A": "(14,27)", "B": "(16,30)", "C": "(18,33)", "D": "(20,36)"}, "correct_answer": "D"},
    {"id": 58, "difficulty": "medium", "question": "Construct division of 23 cm segment in ratio 7:9. Distance from division point to point that divides same segment in ratio 9:7:", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "A(3,8), B(9,20). Point dividing in ratio 2:5:", "options": {"A": "(4.2,11.2)", "B": "(4.7,12.6)", "C": "(5.3,14)", "D": "(5.8,15.4)"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "Divide 31.5 cm segment in ratio 2:3:4. Product of lengths of two larger parts:", "options": {"A": "147 cm²", "B": "162 cm²", "C": "189 cm²", "D": "216 cm²"}, "correct_answer": "C"},
    {"id": 61, "difficulty": "medium", "question": "P(-4,6), Q(8,-6). Point dividing in ratio 1:1:", "options": {"A": "(0,0)", "B": "(2,0)", "C": "(4,0)", "D": "(6,0)"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "medium", "question": "Construct division of 26 cm segment in ratio 8:9. Then divide each part in ratio 1:2 and 2:1 respectively. Distance between these new points:", "options": {"A": "13/3 cm", "B": "26/3 cm", "C": "13 cm", "D": "26 cm"}, "correct_answer": "B"},
    {"id": 63, "difficulty": "medium", "question": "A(1,4), B(7,22). Point dividing in ratio k:1 is (4,13). Find k.", "options": {"A": "0.5", "B": "1", "C": "1.5", "D": "2"}, "correct_answer": "C"},
    {"id": 64, "difficulty": "medium", "question": "Divide 33.6 cm segment in ratio 5:7. Sum of reciprocal of parts:", "options": {"A": "1/2.8", "B": "1/2.4", "C": "1/2.0", "D": "1/1.8"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "P(0,5), Q(12,17). R divides PQ in ratio 3:4. Find area of triangle with vertices P,Q,R.", "options": {"A": "12", "B": "18", "C": "24", "D": "30"}, "correct_answer": "B"},
    {"id": 66, "difficulty": "medium", "question": "Construct division of 29 cm segment in ratio 10:11. Point dividing the segment joining division point to midpoint in ratio 3:2 is how far from nearest end?", "options": {"A": "8 cm", "B": "9 cm", "C": "10 cm", "D": "11 cm"}, "correct_answer": "C"},
    {"id": 67, "difficulty": "medium", "question": "A(2,2), B(14,26). Points dividing in ratio 1:2 and 2:1. Distance between them:", "options": {"A": "4√5", "B": "6√5", "C": "8√5", "D": "10√5"}, "correct_answer": "C"},
    {"id": 68, "difficulty": "medium", "question": "Divide 36.4 cm segment in ratio 3:4:6. Length of middle part as percentage of whole:", "options": {"A": "20%", "B": "25%", "C": "30%", "D": "35%"}, "correct_answer": "C"},
    {"id": 69, "difficulty": "medium", "question": "P(-3,2), Q(9,14). R divides PQ externally in ratio 5:3. Find R.", "options": {"A": "(15,20)", "B": "(18,23)", "C": "(21,26)", "D": "(24,29)"}, "correct_answer": "D"},
    {"id": 70, "difficulty": "medium", "question": "Construct division of 32 cm segment in ratio 9:11. Point that divides line joining division point to an end in ratio equal to original ratio is how far from that end?", "options": {"A": "12 cm", "B": "14.4 cm", "C": "16.8 cm", "D": "19.2 cm"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "A(4,9), B(10,21). Point dividing in ratio 4:7:", "options": {"A": "(5.8,13.2)", "B": "(6.4,14.4)", "C": "(7.0,15.6)", "D": "(7.6,16.8)"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Divide 40 cm segment in ratio 3:5:7. Sum of two smaller parts:", "options": {"A": "16 cm", "B": "20 cm", "C": "24 cm", "D": "28 cm"}, "correct_answer": "C"},
    {"id": 73, "difficulty": "medium", "question": "P(1,3), Q(7,15). R divides PQ such that PR = 2QR. Find R.", "options": {"A": "(3,7)", "B": "(4,9)", "C": "(5,11)", "D": "(6,13)"}, "correct_answer": "C"},
    {"id": 74, "difficulty": "medium", "question": "Construct division of 35 cm segment in ratio 11:12. Distance from division point to point dividing whole segment in ratio equal to reciprocal of original:", "options": {"A": "2.5 cm", "B": "5 cm", "C": "7.5 cm", "D": "10 cm"}, "correct_answer": "B"},
    {"id": 75, "difficulty": "medium", "question": "A(-2,5), B(6,13). Midpoint of segment joining points dividing AB in ratios 1:4 and 4:1:", "options": {"A": "(0,7)", "B": "(1,8)", "C": "(2,9)", "D": "(3,10)"}, "correct_answer": "C"},
    {"id": 76, "difficulty": "medium", "question": "Divide 42 cm segment in ratio 4:5:9. Ratio of largest to smallest part:", "options": {"A": "9:4", "B": "9:5", "C": "5:4", "D": "4:9"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "P(0,0), Q(15,20). R divides PQ in ratio 3:2. Find distance QR.", "options": {"A": "5√13", "B": "10", "C": "10√2", "D": "15"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "Construct division of 38 cm segment in ratio 13:14. Divide the larger part in original ratio. Distance from this new point to nearest end:", "options": {"A": "13 cm", "B": "14 cm", "C": "15 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 79, "difficulty": "medium", "question": "A(3,6), B(12,24). Point dividing in ratio 5:8:", "options": {"A": "(5.3,10.6)", "B": "(6.2,12.4)", "C": "(7.1,14.2)", "D": "(8.0,16.0)"}, "correct_answer": "B"},
    {"id": 80, "difficulty": "medium", "question": "Divide 45.5 cm segment in ratio 6:7. Difference between parts as percentage of whole:", "options": {"A": "7.7%", "B": "8.3%", "C": "9.1%", "D": "10.0%"}, "correct_answer": "A"},

    {"id": 81, "difficulty": "hard", "question": "Construct division of line segment AB=12 cm in ratio 3:5. Construct point P on AB such that AP:PB = 3:5 and point Q on BA produced such that AQ:QB = 3:5. Find PQ.", "options": {"A": "18 cm", "B": "21 cm", "C": "24 cm", "D": "27 cm"}, "correct_answer": "C"},
    {"id": 82, "difficulty": "hard", "question": "A line segment of length 15 cm is divided in ratio 2:3:4. Construct circles with centers at division points and radii equal to lengths of adjacent parts. Find sum of areas of two smaller circles if they touch internally.", "options": {"A": "13π cm²", "B": "17π cm²", "C": "21π cm²", "D": "25π cm²"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Construct triangle with vertices A(0,0), B(12,0), C(0,16). Construct points dividing AB, BC, CA in ratio 3:1. Find area of triangle formed by these points.", "options": {"A": "18 sq units", "B": "24 sq units", "C": "30 sq units", "D": "36 sq units"}, "correct_answer": "B"},
    {"id": 84, "difficulty": "hard", "question": "Divide segment of length 20 cm in ratio 3:7. On the larger part as diameter, construct semicircle. On the whole segment as base, construct equilateral triangle. Find distance from center of semicircle to apex of triangle.", "options": {"A": "5√7 cm", "B": "10√3/3 cm", "C": "5√13 cm", "D": "10√2 cm"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Construct points A(1,1), B(7,3), C(4,7). Construct point D dividing AB in ratio 2:1 and point E dividing AC in ratio 1:2. Find area of quadrilateral BCDE.", "options": {"A": "10 sq units", "B": "12 sq units", "C": "14 sq units", "D": "16 sq units"}, "correct_answer": "C"},
    {"id": 86, "difficulty": "hard", "question": "Divide segment of length 24 cm in ratio 1:2:3. Construct circles with centers at division points with radii in arithmetic progression starting with 1 cm. Find radius of circle touching all three circles externally.", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "B"},
    {"id": 87, "difficulty": "hard", "question": "Construct triangle with sides 10 cm, 17 cm, 21 cm. Construct points dividing each side in ratio equal to ratio of other two sides. Show these three points are collinear and find ratio in which they divide each other.", "options": {"A": "10:21", "B": "17:21", "C": "10:17", "D": "They are equal"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Divide segment of length 18 cm in golden ratio (approximately 1:1.618). Construct square on smaller part and equilateral triangle on larger part. Find ratio of area of square to area of triangle.", "options": {"A": "√3:2", "B": "2:√3", "C": "√5:2", "D": "2:√5"}, "correct_answer": "B"},
    {"id": 89, "difficulty": "hard", "question": "Construct quadrilateral A(0,0), B(8,0), C(10,6), D(2,6). Construct points dividing AB, BC, CD, DA in ratio 3:5. Find area of quadrilateral formed by these points.", "options": {"A": "15 sq units", "B": "18 sq units", "C": "21 sq units", "D": "24 sq units"}, "correct_answer": "B"},
    {"id": 90, "difficulty": "hard", "question": "Divide segment of length 30 cm in ratio 2:3. On each part, construct semicircle on same side. Construct circle touching both semicircles internally. Find its radius.", "options": {"A": "3.6 cm", "B": "4.5 cm", "C": "5.4 cm", "D": "6.3 cm"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "Construct triangle with vertices (0,0), (12,0), (0,9). Construct points dividing sides in ratios such that lines joining vertices to division points on opposite sides are concurrent. If division ratios are 2:1, 3:2, k:1, find k.", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "B"},
    {"id": 92, "difficulty": "hard", "question": "Divide segment of length 25 cm in ratio 3:4:5. Construct equilateral triangles on each part as bases, all on same side. Find distance between apex of first and third triangles.", "options": {"A": "5√21 cm", "B": "5√23 cm", "C": "5√25 cm", "D": "5√27 cm"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "Construct regular hexagon of side 6 cm. Construct points dividing each side in ratio 1:2 alternately. Find perimeter of hexagon formed by these points.", "options": {"A": "12√3 cm", "B": "18 cm", "C": "18√3 cm", "D": "24 cm"}, "correct_answer": "B"},
    {"id": 94, "difficulty": "hard", "question": "Divide segment of length 20 cm in ratio a:b such that when squares are constructed on both parts, difference of areas is 96 cm². Construct this division and find a:b.", "options": {"A": "1:2", "B": "2:3", "C": "3:4", "D": "4:5"}, "correct_answer": "B"},
    {"id": 95, "difficulty": "hard", "question": "Construct triangle ABC with AB=14 cm, BC=15 cm, CA=13 cm. Construct points D,E,F on BC, CA, AB respectively such that BD:DC = CE:EA = AF:FB = 2:3. Find ratio of area of triangle DEF to area of triangle ABC.", "options": {"A": "4:25", "B": "6:25", "C": "9:25", "D": "12:25"}, "correct_answer": "B"},
    {"id": 96, "difficulty": "hard", "question": "Divide segment of length 28 cm in ratio 3:5. On whole segment as diameter, construct semicircle. On the two parts as diameters, construct semicircles on opposite sides. Find area of region between the three semicircles.", "options": {"A": "42π cm²", "B": "48π cm²", "C": "54π cm²", "D": "60π cm²"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Construct quadrilateral with vertices (0,0), (10,0), (12,8), (2,8). Construct points dividing sides in ratio equal to ratio of adjacent sides. Show these points are concyclic and find radius of circle.", "options": {"A": "5", "B": "√41", "C": "√61", "D": "√89"}, "correct_answer": "C"},
    {"id": 98, "difficulty": "hard", "question": "Divide segment of length 32 cm in golden ratio. Construct regular pentagon on smaller part and decagon on larger part. Find ratio of their perimeters.", "options": {"A": "1:φ", "B": "φ:1", "C": "1:φ²", "D": "φ²:1"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Construct triangle with sides 18 cm, 24 cm, 30 cm. Construct points dividing sides in ratio equal to ratio of sines of opposite angles. Find area of triangle formed by these points.", "options": {"A": "36 cm²", "B": "54 cm²", "C": "72 cm²", "D": "108 cm²"}, "correct_answer": "B"},
    {"id": 100, "difficulty": "hard", "question": "Divide segment of length 40 cm in ratio 1:√2. Construct square on smaller part and circle with larger part as diameter. Find ratio of area of square to area of circle.", "options": {"A": "1:π", "B": "2:π", "C": "√2:π", "D": "1:π√2"}, "correct_answer": "B"}
  ]
}