{
  "title": "Similarity of Triangles - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Two triangles are similar if their corresponding angles are:", "options": {"A": "complementary", "B": "supplementary", "C": "equal", "D": "unequal"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "If two triangles are similar, their corresponding sides are:", "options": {"A": "equal", "B": "proportional", "C": "reciprocal", "D": "unrelated"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "Which similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar?", "options": {"A": "SSS", "B": "SAS", "C": "AAA", "D": "ASA"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "If ΔABC ∼ ΔDEF, which of the following is true?", "options": {"A": "AB/DE = BC/EF = CA/FD", "B": "AB/DE = BC/FD = CA/EF", "C": "AB/EF = BC/DE = CA/FD", "D": "AB/FD = BC/DE = CA/EF"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "All equilateral triangles are:", "options": {"A": "congruent", "B": "similar", "C": "different", "D": "right-angled"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If two triangles are similar, the ratio of their areas is equal to:", "options": {"A": "ratio of corresponding sides", "B": "square of ratio of corresponding sides", "C": "cube of ratio of corresponding sides", "D": "square root of ratio of corresponding sides"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "In two similar triangles, if the ratio of corresponding sides is 2:3, then the ratio of their areas is:", "options": {"A": "2:3", "B": "3:2", "C": "4:9", "D": "9:4"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "Which similarity criterion involves two sides and the included angle?", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "ASA"}, "correct_answer": "C"},
    {"id": 9, "difficulty": "easy", "question": "If in ΔABC and ΔDEF, ∠A = ∠D and ∠B = ∠E, then:", "options": {"A": "ΔABC ≅ ΔDEF", "B": "ΔABC ∼ ΔDEF", "C": "ΔABC ≠ ΔDEF", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "In two similar triangles, the ratio of corresponding altitudes is equal to:", "options": {"A": "ratio of areas", "B": "ratio of perimeters", "C": "ratio of corresponding sides", "D": "inverse ratio of sides"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "If ΔABC ∼ ΔPQR and AB = 4 cm, PQ = 6 cm, then the scale factor is:", "options": {"A": "2:3", "B": "3:2", "C": "4:6", "D": "6:4"}, "correct_answer": "A"},
    {"id": 12, "difficulty": "easy", "question": "All circles are:", "options": {"A": "congruent", "B": "similar", "C": "different", "D": "equal"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "If two triangles are similar, their perimeters are in the ratio of:", "options": {"A": "their areas", "B": "their corresponding sides", "C": "square of corresponding sides", "D": "cube of corresponding sides"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "Which similarity criterion involves all three sides?", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "ASA"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "If ΔABC ∼ ΔDEF and AB = 3 cm, DE = 5 cm, BC = 6 cm, then EF = ?", "options": {"A": "3.6 cm", "B": "10 cm", "C": "9 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "All squares are:", "options": {"A": "congruent", "B": "similar", "C": "different", "D": "equal"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "If the ratio of corresponding sides of two similar triangles is 3:4, the ratio of their perimeters is:", "options": {"A": "3:4", "B": "4:3", "C": "9:16", "D": "16:9"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "In ΔABC, DE || BC. If AD = 3 cm, DB = 6 cm, AE = 4 cm, then EC = ?", "options": {"A": "2 cm", "B": "4 cm", "C": "6 cm", "D": "8 cm"}, "correct_answer": "D"},
    {"id": 19, "difficulty": "easy", "question": "If two triangles are congruent, they are always:", "options": {"A": "similar", "B": "different", "C": "equal in area only", "D": "not similar"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "The scale factor of two similar triangles is 2:5. If a side of the smaller triangle is 8 cm, the corresponding side of larger triangle is:", "options": {"A": "16 cm", "B": "20 cm", "C": "25 cm", "D": "40 cm"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 2 cm, DB = 3 cm, and AE = 3 cm, find AC.", "options": {"A": "4.5 cm", "B": "5 cm", "C": "6 cm", "D": "7.5 cm"}, "correct_answer": "D"},
    {"id": 22, "difficulty": "medium", "question": "ΔABC ∼ ΔDEF. If AB = 4 cm, DE = 6 cm, and area of ΔABC = 16 cm², find area of ΔDEF.", "options": {"A": "24 cm²", "B": "32 cm²", "C": "36 cm²", "D": "48 cm²"}, "correct_answer": "C"},
    {"id": 23, "difficulty": "medium", "question": "In ΔABC, ∠B = 90°. BD ⟂ AC. If AD = 4 cm, DC = 9 cm, find BD.", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "The perimeters of two similar triangles are 25 cm and 15 cm. If one side of first triangle is 9 cm, find the corresponding side of second triangle.", "options": {"A": "3.6 cm", "B": "5.4 cm", "C": "6.5 cm", "D": "7.5 cm"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD/DB = 3/5 and AC = 4.8 cm, find AE.", "options": {"A": "1.8 cm", "B": "2.4 cm", "C": "3 cm", "D": "3.6 cm"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Two similar triangles have areas 81 cm² and 144 cm². If a side of smaller triangle is 9 cm, find the corresponding side of larger triangle.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = x-3, DB = 3x-19, AE = x-4, EC = 3x-21, find x.", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "C"},
    {"id": 28, "difficulty": "medium", "question": "ΔABC ∼ ΔPQR. If AB = 5 cm, BC = 6 cm, CA = 7 cm, and PQ = 10 cm, find PR.", "options": {"A": "12 cm", "B": "14 cm", "C": "16 cm", "D": "18 cm"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "The areas of two similar triangles are 100 cm² and 64 cm². If a median of larger triangle is 15 cm, find the corresponding median of smaller triangle.", "options": {"A": "9.6 cm", "B": "10 cm", "C": "12 cm", "D": "13.5 cm"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "In ΔABC, ∠B = 90°, BD ⟂ AC. If AB = 6 cm, BC = 8 cm, find BD.", "options": {"A": "3.6 cm", "B": "4.2 cm", "C": "4.8 cm", "D": "5.4 cm"}, "correct_answer": "C"},
    {"id": 31, "difficulty": "medium", "question": "Two poles of heights 6 m and 11 m stand vertically on ground. If the distance between their feet is 12 m, find the distance between their tops.", "options": {"A": "13 m", "B": "14 m", "C": "15 m", "D": "16 m"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "ΔABC ∼ ΔDEF. If AB = 3 cm, BC = 4 cm, CA = 5 cm, and the perimeter of ΔDEF is 36 cm, find EF.", "options": {"A": "8 cm", "B": "9 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "D"},
    {"id": 33, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 2 cm, AB = 8 cm, and AE = 3 cm, find EC.", "options": {"A": "6 cm", "B": "8 cm", "C": "9 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 34, "difficulty": "medium", "question": "The areas of two similar triangles are 121 cm² and 81 cm². If the perimeter of larger triangle is 44 cm, find the perimeter of smaller triangle.", "options": {"A": "30 cm", "B": "32 cm", "C": "34 cm", "D": "36 cm"}, "correct_answer": "D"},
    {"id": 35, "difficulty": "medium", "question": "In ΔABC, ∠A = 90°, AD ⟂ BC. If BD = 4 cm, DC = 9 cm, find AD.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 36, "difficulty": "medium", "question": "ΔABC ∼ ΔPQR. If AB = 6 cm, BC = 8 cm, CA = 10 cm, and the smallest side of ΔPQR is 9 cm, find its largest side.", "options": {"A": "12 cm", "B": "15 cm", "C": "18 cm", "D": "21 cm"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "In trapezium ABCD, AB || DC. If diagonals AC and BD intersect at O, and AB = 6 cm, DC = 9 cm, find AO/OC.", "options": {"A": "2/3", "B": "3/2", "C": "2/5", "D": "5/2"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "Two similar triangles have their corresponding sides in ratio 5:3. If the area of larger triangle is 125 cm², find the area of smaller triangle.", "options": {"A": "45 cm²", "B": "50 cm²", "C": "55 cm²", "D": "60 cm²"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 4 cm, AE = 6 cm, EC = 9 cm, find DB.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "8 cm"}, "correct_answer": "C"},
    {"id": 40, "difficulty": "medium", "question": "ΔABC ∼ ΔDEF. If the ratio of their perimeters is 4:9, and area of ΔABC is 64 cm², find area of ΔDEF.", "options": {"A": "144 cm²", "B": "196 cm²", "C": "256 cm²", "D": "324 cm²"}, "correct_answer": "D"},
    {"id": 41, "difficulty": "medium", "question": "In ΔABC, ∠B = 90°. BD ⟂ AC. If AD = 2 cm, DC = 8 cm, find AB.", "options": {"A": "2√5 cm", "B": "4 cm", "C": "2√10 cm", "D": "6 cm"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "medium", "question": "The perimeters of two similar triangles are 30 cm and 45 cm. If one side of first triangle is 12 cm, find the corresponding side of second triangle.", "options": {"A": "15 cm", "B": "16 cm", "C": "18 cm", "D": "20 cm"}, "correct_answer": "C"},
    {"id": 43, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 2x, DB = x+4, AE = 3x, EC = 2x+6, find x.", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "medium", "question": "Two similar triangles have areas 49 cm² and 64 cm². The sum of a pair of corresponding sides is 15 cm. Find these sides.", "options": {"A": "7 cm, 8 cm", "B": "8 cm, 7 cm", "C": "6 cm, 9 cm", "D": "5 cm, 10 cm"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "medium", "question": "In ΔABC, ∠A = 90°, AD ⟂ BC. If AB = 8 cm, AC = 15 cm, find BD.", "options": {"A": "64/17 cm", "B": "128/17 cm", "C": "64/15 cm", "D": "128/15 cm"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "medium", "question": "ΔABC ∼ ΔDEF. If AB = 2 cm, BC = 3 cm, CA = 4 cm, and the perimeter of ΔDEF is 27 cm, find its sides.", "options": {"A": "6 cm, 9 cm, 12 cm", "B": "5 cm, 8 cm, 14 cm", "C": "4 cm, 10 cm, 13 cm", "D": "7 cm, 8 cm, 12 cm"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "medium", "question": "In trapezium ABCD, AB || DC. Diagonals intersect at O. If AB = 4 cm, DC = 10 cm, and area of ΔAOB = 16 cm², find area of ΔCOD.", "options": {"A": "25 cm²", "B": "36 cm²", "C": "49 cm²", "D": "100 cm²"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "medium", "question": "Two similar triangles have perimeters 18 cm and 24 cm. If area of smaller triangle is 36 cm², find area of larger triangle.", "options": {"A": "48 cm²", "B": "54 cm²", "C": "64 cm²", "D": "72 cm²"}, "correct_answer": "C"},
    {"id": 49, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 6 cm, DB = 9 cm, and area of ΔADE = 24 cm², find area of trapezium DBCE.", "options": {"A": "54 cm²", "B": "75 cm²", "C": "96 cm²", "D": "120 cm²"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "medium", "question": "ΔABC ∼ ΔPQR. If AB = 5 cm, BC = 7 cm, CA = 9 cm, and the longest side of ΔPQR is 18 cm, find its shortest side.", "options": {"A": "8 cm", "B": "9 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "C"},
    {"id": 51, "difficulty": "medium", "question": "In ΔABC, ∠B = 90°. BD ⟂ AC. If AB = 12 cm, BC = 16 cm, find DC.", "options": {"A": "12.8 cm", "B": "14.4 cm", "C": "16 cm", "D": "18.2 cm"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "The areas of two similar triangles are 144 cm² and 225 cm². If a side of larger triangle is 25 cm, find the corresponding side of smaller triangle.", "options": {"A": "16 cm", "B": "18 cm", "C": "20 cm", "D": "22 cm"}, "correct_answer": "C"},
    {"id": 53, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 3 cm, DB = 5 cm, and area of ΔADE = 27 cm², find area of ΔABC.", "options": {"A": "64 cm²", "B": "96 cm²", "C": "128 cm²", "D": "192 cm²"}, "correct_answer": "C"},
    {"id": 54, "difficulty": "medium", "question": "Two poles of heights 8 m and 15 m stand vertically. Distance between their tops is 17 m. Find distance between their feet.", "options": {"A": "10 m", "B": "12 m", "C": "14 m", "D": "15 m"}, "correct_answer": "D"},
    {"id": 55, "difficulty": "medium", "question": "In ΔABC, ∠A = 90°, AD ⟂ BC. If BD = 9 cm, DC = 16 cm, find AB and AC.", "options": {"A": "AB=12, AC=20", "B": "AB=15, AC=20", "C": "AB=12, AC=15", "D": "AB=15, AC=25"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "medium", "question": "ΔABC ∼ ΔDEF. Ratio of their areas is 16:25. If EF = 7.5 cm, find BC.", "options": {"A": "4.8 cm", "B": "5 cm", "C": "6 cm", "D": "7.2 cm"}, "correct_answer": "C"},
    {"id": 57, "difficulty": "medium", "question": "In trapezium ABCD, AB || DC. Diagonals intersect at O. If AB = 3 cm, DC = 6 cm, find area(ΔAOB)/area(ΔCOD).", "options": {"A": "1/4", "B": "1/2", "C": "1/3", "D": "2/3"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "Two similar triangles have perimeters 20 cm and 30 cm. If an altitude of smaller triangle is 8 cm, find the corresponding altitude of larger triangle.", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 4 cm, DB = 6 cm, and area of ΔADE = 20 cm², find area of quadrilateral DBCE.", "options": {"A": "25 cm²", "B": "45 cm²", "C": "55 cm²", "D": "65 cm²"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "medium", "question": "ΔABC ∼ ΔPQR. If AB = 8 cm, BC = 10 cm, CA = 12 cm, and perimeter of ΔPQR = 60 cm, find PQ.", "options": {"A": "12 cm", "B": "14 cm", "C": "16 cm", "D": "18 cm"}, "correct_answer": "C"},
    {"id": 61, "difficulty": "medium", "question": "In ΔABC, ∠B = 90°. BD ⟂ AC. If AD = 3 cm, AC = 12 cm, find AB.", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "C"},
    {"id": 62, "difficulty": "medium", "question": "The areas of two similar triangles are 81 cm² and 121 cm². If an altitude of first triangle is 9 cm, find the corresponding altitude of second triangle.", "options": {"A": "10 cm", "B": "11 cm", "C": "12 cm", "D": "13 cm"}, "correct_answer": "B"},
    {"id": 63, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 2 cm, AB = 10 cm, and area of ΔABC = 100 cm², find area of ΔADE.", "options": {"A": "4 cm²", "B": "8 cm²", "C": "12 cm²", "D": "16 cm²"}, "correct_answer": "A"},
    {"id": 64, "difficulty": "medium", "question": "Two similar triangles have corresponding sides in ratio 3:7. If area of smaller triangle is 54 cm², find area of larger triangle.", "options": {"A": "126 cm²", "B": "196 cm²", "C": "252 cm²", "D": "294 cm²"}, "correct_answer": "D"},
    {"id": 65, "difficulty": "medium", "question": "In ΔABC, ∠A = 90°, AD ⟂ BC. If BD = 6 cm, DC = 10 cm, find AD.", "options": {"A": "2√15 cm", "B": "4√5 cm", "C": "2√30 cm", "D": "4√10 cm"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "ΔABC ∼ ΔDEF. Ratio of their altitudes is 4:5. If area of ΔABC is 64 cm², find area of ΔDEF.", "options": {"A": "80 cm²", "B": "100 cm²", "C": "125 cm²", "D": "150 cm²"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "In trapezium ABCD, AB || DC. If AB = 5 cm, DC = 10 cm, and area of ΔAOB = 20 cm², find area of trapezium ABCD.", "options": {"A": "60 cm²", "B": "75 cm²", "C": "90 cm²", "D": "120 cm²"}, "correct_answer": "C"},
    {"id": 68, "difficulty": "medium", "question": "Two similar triangles have perimeters 24 cm and 36 cm. If area of larger triangle is 108 cm², find area of smaller triangle.", "options": {"A": "48 cm²", "B": "54 cm²", "C": "64 cm²", "D": "72 cm²"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 5 cm, DB = 10 cm, and area of trapezium DBCE = 105 cm², find area of ΔADE.", "options": {"A": "15 cm²", "B": "20 cm²", "C": "25 cm²", "D": "30 cm²"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "medium", "question": "ΔABC ∼ ΔPQR. If AB = 6 cm, BC = 8 cm, AC = 10 cm, and the median to the longest side of ΔPQR is 12 cm, find the corresponding median of ΔABC.", "options": {"A": "4.8 cm", "B": "6 cm", "C": "7.2 cm", "D": "9.6 cm"}, "correct_answer": "C"},
    {"id": 71, "difficulty": "medium", "question": "In ΔABC, ∠B = 90°. BD ⟂ AC. If AB = 9 cm, BC = 12 cm, find the ratio AD:DC.", "options": {"A": "9:16", "B": "16:9", "C": "3:4", "D": "4:3"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "The areas of two similar triangles are 100 cm² and 400 cm². If the perimeter of smaller triangle is 25 cm, find the perimeter of larger triangle.", "options": {"A": "50 cm", "B": "75 cm", "C": "100 cm", "D": "125 cm"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 3 cm, AE = 4 cm, EC = 8 cm, find the ratio area(ΔADE):area(trapezium DBCE).", "options": {"A": "1:3", "B": "1:4", "C": "1:5", "D": "1:8"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "Two similar triangles have corresponding medians in ratio 5:8. If area of smaller triangle is 75 cm², find area of larger triangle.", "options": {"A": "120 cm²", "B": "150 cm²", "C": "192 cm²", "D": "240 cm²"}, "correct_answer": "C"},
    {"id": 75, "difficulty": "medium", "question": "In ΔABC, ∠A = 90°, AD ⟂ BC. If AB = 6 cm, AC = 8 cm, find the length of AD.", "options": {"A": "4.8 cm", "B": "5.2 cm", "C": "5.6 cm", "D": "6.4 cm"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "ΔABC ∼ ΔDEF. If the ratio of their perimeters is 2:3, and area of ΔDEF is 54 cm², find area of ΔABC.", "options": {"A": "24 cm²", "B": "30 cm²", "C": "36 cm²", "D": "42 cm²"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "In trapezium ABCD, AB || DC. Diagonals intersect at O. If AB = 8 cm, DC = 12 cm, find the ratio of areas of ΔAOD and ΔBOC.", "options": {"A": "2:3", "B": "3:2", "C": "4:9", "D": "9:4"}, "correct_answer": "C"},
    {"id": 78, "difficulty": "medium", "question": "Two similar triangles have areas 36 cm² and 100 cm². If a side of smaller triangle is 4.5 cm, find the corresponding side of larger triangle.", "options": {"A": "6.5 cm", "B": "7.5 cm", "C": "8.5 cm", "D": "9.5 cm"}, "correct_answer": "B"},
    {"id": 79, "difficulty": "medium", "question": "In ΔABC, DE || BC. If AD = 2 cm, DB = 3 cm, and area of ΔABC = 50 cm², find area of ΔADE.", "options": {"A": "8 cm²", "B": "10 cm²", "C": "12 cm²", "D": "15 cm²"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "ΔABC ∼ ΔPQR. If AB = 7 cm, BC = 9 cm, CA = 11 cm, and the shortest side of ΔPQR is 14 cm, find its longest side.", "options": {"A": "18 cm", "B": "20 cm", "C": "22 cm", "D": "24 cm"}, "correct_answer": "C"},
    {"id": 81, "difficulty": "hard", "question": "In ΔABC, DE || BC. If AD = 1 cm, DB = 2 cm, and area of ΔADE = 3 cm², find area of quadrilateral DBCE.", "options": {"A": "15 cm²", "B": "18 cm²", "C": "21 cm²", "D": "24 cm²"}, "correct_answer": "D"},
    {"id": 82, "difficulty": "hard", "question": "The areas of two similar triangles are in ratio 9:25. If a median of smaller triangle is 6 cm, find the corresponding median of larger triangle.", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "15 cm"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "In ΔABC, ∠B = 90°. BD ⟂ AC. If AD = 4 cm, DC = 5 cm, find the ratio AB:BC.", "options": {"A": "2:√5", "B": "√5:2", "C": "2:3", "D": "3:2"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Two poles of heights a and b stand apart at distance d. The line joining their tops makes angle θ with horizontal. Prove that d = (b-a) cot θ.", "options": {"A": "Always true", "B": "Sometimes true", "C": "Never true", "D": "Cannot be determined"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "In ΔABC, D and E are points on AB and AC such that DE || BC. If AD = 2 cm, AB = 6 cm, and area of ΔADE = 8 cm², find area of trapezium DBCE.", "options": {"A": "40 cm²", "B": "48 cm²", "C": "56 cm²", "D": "64 cm²"}, "correct_answer": "D"},
    {"id": 86, "difficulty": "hard", "question": "The perimeters of two similar triangles are 30 cm and 40 cm. If an altitude of first triangle is 9 cm, find the sum of corresponding altitude and median of second triangle if median is 2 cm more than altitude.", "options": {"A": "15 cm", "B": "17 cm", "C": "19 cm", "D": "21 cm"}, "correct_answer": "C"},
    {"id": 87, "difficulty": "hard", "question": "In trapezium ABCD, AB || DC. Diagonals intersect at O. If AB = 6 cm, DC = 10 cm, and area of ΔAOD = 18 cm², find area of trapezium ABCD.", "options": {"A": "96 cm²", "B": "108 cm²", "C": "120 cm²", "D": "128 cm²"}, "correct_answer": "D"},
    {"id": 88, "difficulty": "hard", "question": "ΔABC ∼ ΔDEF. Area of ΔABC is 16 cm² and area of ΔDEF is 49 cm². If BC = 4 cm, find the length of EF.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "C"},
    {"id": 89, "difficulty": "hard", "question": "In ΔABC, ∠A = 90°. AD ⟂ BC. If BD:DC = 1:3 and AD = 6 cm, find AB and AC.", "options": {"A": "AB=3√2, AC=6√2", "B": "AB=4√3, AC=8√3", "C": "AB=6, AC=12", "D": "AB=2√10, AC=4√10"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "Two similar triangles have their corresponding sides in ratio k:1. If the difference of their areas is m times the area of smaller triangle, find m in terms of k.", "options": {"A": "k²-1", "B": "k-1", "C": "k²+1", "D": "k+1"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "In ΔABC, DE || BC. If AD = 1 cm, DB = 2 cm, and the ratio of areas of ΔADE and trapezium DBCE is 1:8, find area of ΔABC.", "options": {"A": "9 cm²", "B": "12 cm²", "C": "15 cm²", "D": "18 cm²"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "The areas of two similar triangles are 81 cm² and 144 cm². If the sum of a pair of corresponding sides is 25 cm, find these sides.", "options": {"A": "9 cm, 16 cm", "B": "10 cm, 15 cm", "C": "11 cm, 14 cm", "D": "12 cm, 13 cm"}, "correct_answer": "B"},
    {"id": 93, "difficulty": "hard", "question": "In ΔABC, ∠B = 90°. BD ⟂ AC. If AB = 8 cm, BC = 15 cm, find the lengths of AD, BD, and DC.", "options": {"A": "AD=64/17, BD=120/17, DC=225/17", "B": "AD=8/3, BD=40/3, DC=25", "C": "AD=4, BD=12, DC=9", "D": "AD=6.4, BD=9.6, DC=14.4"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Two triangles are similar. The ratio of their areas is 4:9. If the height of the smaller triangle is 6 cm, find the corresponding height of larger triangle. Also, if base of larger triangle is 15 cm, find base of smaller triangle.", "options": {"A": "Height=9 cm, Base=10 cm", "B": "Height=8 cm, Base=12 cm", "C": "Height=9 cm, Base=12 cm", "D": "Height=8 cm, Base=10 cm"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "In trapezium ABCD, AB || DC. Diagonals intersect at O. If AB = 4 cm, DC = 9 cm, and area of ΔAOB = 16 cm², find the ratio of areas of ΔAOD and ΔBOC.", "options": {"A": "4:9", "B": "9:4", "C": "2:3", "D": "3:2"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "ΔABC ∼ ΔPQR. If AD and PS are medians of ΔABC and ΔPQR respectively, prove that AB/PQ = AD/PS. If AB = 6 cm, PQ = 9 cm, and AD = 4 cm, find PS.", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 97, "difficulty": "hard", "question": "Two similar triangles have perimeters 24 cm and 36 cm. If the area of the smaller triangle is 48 cm², find the difference between the areas of the two triangles.", "options": {"A": "60 cm²", "B": "72 cm²", "C": "84 cm²", "D": "96 cm²"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "In ΔABC, DE || BC. If AD = x, DB = x+2, AE = x-1, EC = x+3, find x and the ratio AD:DB.", "options": {"A": "x=3, AD:DB=3:5", "B": "x=4, AD:DB=2:3", "C": "x=5, AD:DB=5:7", "D": "x=6, AD:DB=3:4"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "The areas of two similar triangles are 100 cm² and 400 cm². If the perimeter of the larger triangle is 50 cm, find the perimeter of the smaller triangle and the ratio of their corresponding medians.", "options": {"A": "25 cm, 1:2", "B": "30 cm, 1:√2", "C": "35 cm, 1:3", "D": "40 cm, 1:4"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "In ΔABC, ∠A = 90°. AD ⟂ BC. Prove that AB² = BD × BC and AC² = CD × BC. If BD = 4 cm, DC = 9 cm, find AB, AC, and AD.", "options": {"A": "AB=6, AC=13, AD=√52", "B": "AB=2√13, AC=3√13, AD=6", "C": "AB=√52, AC=√117, AD=6", "D": "AB=√52, AC=√117, AD=√36"}, "correct_answer": "C"}
  ]
}