{
"title": "Areas of Similar Triangles - Grade 10 ICSE",
"total_questions": 60,
"questions": [
{"id": 1, "difficulty": "easy", "question": "If two triangles are similar, their areas are proportional to the square of their corresponding sides. True or False? 📐", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Only for right triangles"}, "correct_answer": "A"},
{"id": 2, "difficulty": "easy", "question": "Two similar triangles have sides in ratio 2:3. What is the ratio of their areas? 🧮", "options": {"A": "2:3", "B": "4:9", "C": "3:2", "D": "9:4"}, "correct_answer": "B"},
{"id": 3, "difficulty": "easy", "question": "If the area of a small similar triangle is 16 cm² and the sides are in ratio 1:4 with a larger triangle, what is the area of the larger triangle? ✨", "options": {"A": "64 cm²", "B": "256 cm²", "C": "128 cm²", "D": "32 cm²"}, "correct_answer": "B"},
{"id": 4, "difficulty": "easy", "question": "The ratio of areas of two similar triangles is 9:25. What is the ratio of their corresponding sides? 🔍", "options": {"A": "3:5", "B": "9:25", "C": "5:3", "D": "25:9"}, "correct_answer": "A"},
{"id": 5, "difficulty": "easy", "question": "If two triangles are similar with scale factor k, then area ratio is? ⚖️", "options": {"A": "k", "B": "k²", "C": "2k", "D": "k/2"}, "correct_answer": "B"},
{"id": 6, "difficulty": "easy", "question": "Two similar triangles have areas 36 cm² and 64 cm². What is the ratio of a pair of corresponding sides? 📏", "options": {"A": "6:8", "B": "8:6", "C": "36:64", "D": "3:4"}, "correct_answer": "D"},
{"id": 7, "difficulty": "easy", "question": "Triangle ABC ~ Triangle PQR. If AB = 6 cm and PQ = 9 cm, and area of ABC = 12 cm², find area of PQR. 🔺", "options": {"A": "18 cm²", "B": "27 cm²", "C": "24 cm²", "D": "36 cm²"}, "correct_answer": "B"},
{"id": 8, "difficulty": "easy", "question": "The sides of two similar triangles are in the ratio 5:7. The area of the smaller is 50 cm². What is the area of the larger? 📈", "options": {"A": "70 cm²", "B": "98 cm²", "C": "125 cm²", "D": "245 cm²"}, "correct_answer": "B"},
{"id": 9, "difficulty": "easy", "question": "If the area ratio of two similar triangles is 1:16, what is the ratio of their perimeters? 🧩", "options": {"A": "1:4", "B": "1:16", "C": "4:1", "D": "16:1"}, "correct_answer": "A"},
{"id": 10, "difficulty": "easy", "question": "Two triangles are similar. If a side of the first is 3 times the corresponding side of the second, how many times larger is its area? ✏️", "options": {"A": "3 times", "B": "6 times", "C": "9 times", "D": "12 times"}, "correct_answer": "C"},
{"id": 11, "difficulty": "easy", "question": "Area of Triangle A = 25 cm². Triangle B is similar with sides 1.5 times those of A. What is the area of B? 🔢", "options": {"A": "37.5 cm²", "B": "56.25 cm²", "C": "50 cm²", "D": "75 cm²"}, "correct_answer": "B"},
{"id": 12, "difficulty": "easy", "question": "If the ratio of corresponding altitudes of two similar triangles is 2:5, what is the ratio of their areas? 🏔️", "options": {"A": "2:5", "B": "4:25", "C": "5:2", "D": "25:4"}, "correct_answer": "B"},
{"id": 13, "difficulty": "easy", "question": "Two similar triangles have perimeters in ratio 2:3. What is the ratio of their areas? 📐", "options": {"A": "2:3", "B": "4:9", "C": "8:27", "D": "√2:√3"}, "correct_answer": "B"},
{"id": 14, "difficulty": "easy", "question": "If ΔABC ~ ΔDEF and area(ΔABC)=81 cm², area(ΔDEF)=144 cm², find AB:DE. ➗", "options": {"A": "9:12", "B": "81:144", "C": "3:4", "D": "4:3"}, "correct_answer": "C"},
{"id": 15, "difficulty": "easy", "question": "The area of two similar triangles are 50 cm² and 98 cm². What is the ratio of a pair of corresponding medians? ⚕️", "options": {"A": "5:7", "B": "25:49", "C": "7:5", "D": "√50:√98"}, "correct_answer": "A"},
{"id": 16, "difficulty": "easy", "question": "If the scale factor of two similar triangles is 1/3, what is the ratio of their areas? 🔄", "options": {"A": "1:3", "B": "1:9", "C": "3:1", "D": "9:1"}, "correct_answer": "B"},
{"id": 17, "difficulty": "easy", "question": "Two similar triangles have corresponding sides 8 cm and 12 cm. The area of the smaller is 32 cm². Find area of larger. 📊", "options": {"A": "48 cm²", "B": "72 cm²", "C": "96 cm²", "D": "108 cm²"}, "correct_answer": "B"},
{"id": 18, "difficulty": "easy", "question": "The ratio of areas is 49:81. What is the ratio of corresponding altitudes? 📏", "options": {"A": "7:9", "B": "49:81", "C": "9:7", "D": "81:49"}, "correct_answer": "A"},
{"id": 19, "difficulty": "easy", "question": "If two triangles are similar and one has twice the side length of the other, its area is ____ times. 🎯", "options": {"A": "2", "B": "4", "C": "8", "D": "√2"}, "correct_answer": "B"},
{"id": 20, "difficulty": "easy", "question": "ΔPQR ~ ΔXYZ. PQ=5 cm, XY=10 cm. If area of ΔPQR=15 cm², find area of ΔXYZ. ✅", "options": {"A": "30 cm²", "B": "45 cm²", "C": "60 cm²", "D": "75 cm²"}, "correct_answer": "C"},
{"id": 21, "difficulty": "medium", "question": "Two similar triangles have areas 100 cm² and 225 cm². If a side of the smaller triangle is 12 cm, find the corresponding side of the larger triangle. 🔍", "options": {"A": "15 cm", "B": "18 cm", "C": "20 cm", "D": "24 cm"}, "correct_answer": "B"},
{"id": 22, "difficulty": "medium", "question": "If the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding medians, True or False? 📏", "options": {"A": "True", "B": "False", "C": "Only for equilateral", "D": "Only if right-angled"}, "correct_answer": "A"},
{"id": 23, "difficulty": "medium", "question": "The areas of two similar triangles are 121 cm² and 64 cm². If the sum of a pair of corresponding sides is 30 cm, find these sides. ➕", "options": {"A": "11 cm and 19 cm", "B": "12 cm and 18 cm", "C": "14 cm and 16 cm", "D": "10 cm and 20 cm"}, "correct_answer": "C"},
{"id": 24, "difficulty": "medium", "question": "ΔABC ~ ΔDEF. Area(ΔABC)=45 cm², DE=8 cm, and EF=12 cm. If AB=6 cm, find area(ΔDEF). 🧮", "options": {"A": "80 cm²", "B": "90 cm²", "C": "100 cm²", "D": "120 cm²"}, "correct_answer": "A"},
{"id": 25, "difficulty": "medium", "question": "In two similar triangles, the ratio of two corresponding sides is 3:5. If the area of the first triangle is 54 cm², find the area of the second. 📐", "options": {"A": "90 cm²", "B": "150 cm²", "C": "250 cm²", "D": "300 cm²"}, "correct_answer": "B"},
{"id": 26, "difficulty": "medium", "question": "If the ratio of the perimeters of two similar triangles is 4:7, what is the ratio of their areas? 🔄", "options": {"A": "4:7", "B": "8:14", "C": "16:49", "D": "2:√7"}, "correct_answer": "C"},
{"id": 27, "difficulty": "medium", "question": "Two similar triangles have corresponding altitudes of 6 cm and 9 cm. If the area of the smaller triangle is 24 cm², find the area of the larger. 🏔️", "options": {"A": "36 cm²", "B": "48 cm²", "C": "54 cm²", "D": "81 cm²"}, "correct_answer": "C"},
{"id": 28, "difficulty": "medium", "question": "The sides of two similar triangles are in the ratio 3:8. The area of the larger triangle is 256 cm². What is the area of the smaller? 📉", "options": {"A": "36 cm²", "B": "48 cm²", "C": "64 cm²", "D": "96 cm²"}, "correct_answer": "A"},
{"id": 29, "difficulty": "medium", "question": "If the ratio of areas of two similar triangles is 25:49, what is the ratio of their corresponding medians? ⚕️", "options": {"A": "5:7", "B": "25:49", "C": "7:5", "D": "√25:√49"}, "correct_answer": "A"},
{"id": 30, "difficulty": "medium", "question": "ΔABC ~ ΔPQR. AB=5 cm, BC=7 cm, CA=8 cm. If the area of ΔABC is 10√3 cm² and PQ=10 cm, find area of ΔPQR. 🔺", "options": {"A": "40√3 cm²", "B": "30√3 cm²", "C": "20√3 cm²", "D": "10√3 cm²"}, "correct_answer": "A"},
{"id": 31, "difficulty": "medium", "question": "Two similar triangles have their areas in the ratio 16:81. If a side of the smaller triangle is 8 cm, find the corresponding side of the larger triangle. 📏", "options": {"A": "13.5 cm", "B": "18 cm", "C": "20 cm", "D": "22.5 cm"}, "correct_answer": "B"},
{"id": 32, "difficulty": "medium", "question": "The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding angle bisectors. True or False? ✨", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Not always"}, "correct_answer": "A"},
{"id": 33, "difficulty": "medium", "question": "If the area of ΔABC is 75 cm² and it is similar to ΔDEF with scale factor 2/3, find area of ΔDEF. 🧩", "options": {"A": "33.33 cm²", "B": "50 cm²", "C": "112.5 cm²", "D": "168.75 cm²"}, "correct_answer": "D"},
{"id": 34, "difficulty": "medium", "question": "Two similar triangles have perimeters 24 cm and 36 cm. If the area of the smaller is 48 cm², find the area of the larger. 📊", "options": {"A": "72 cm²", "B": "96 cm²", "C": "108 cm²", "D": "144 cm²"}, "correct_answer": "C"},
{"id": 35, "difficulty": "medium", "question": "The areas of two similar triangles are 144 cm² and 225 cm². If a median of the smaller triangle is 12 cm, find the corresponding median of the larger. ⚕️", "options": {"A": "15 cm", "B": "18 cm", "C": "20 cm", "D": "25 cm"}, "correct_answer": "A"},
{"id": 36, "difficulty": "medium", "question": "If two triangles are similar and the ratio of their corresponding sides is x:y, then the ratio of their altitudes is? 🏔️", "options": {"A": "x:y", "B": "x²:y²", "C": "√x:√y", "D": "y:x"}, "correct_answer": "A"},
{"id": 37, "difficulty": "medium", "question": "ΔABC ~ ΔDEF. AB=6 cm, DE=9 cm. The altitude from A to BC is 4 cm. Find the altitude from D to EF. 📐", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
{"id": 38, "difficulty": "medium", "question": "Two similar triangles have areas 50 m² and 98 m². If a side of the first is 10 m, find the corresponding side of the second. 🔢", "options": {"A": "12 m", "B": "14 m", "C": "16 m", "D": "18 m"}, "correct_answer": "B"},
{"id": 39, "difficulty": "medium", "question": "If the ratio of corresponding sides of two similar triangles is 5:12, and the area of the smaller is 125 cm², find the area of the larger. 📈", "options": {"A": "300 cm²", "B": "600 cm²", "C": "720 cm²", "D": "900 cm²"}, "correct_answer": "C"},
{"id": 40, "difficulty": "medium", "question": "The ratio of the areas of two similar triangles is 9:25. If an altitude of the smaller triangle is 6 cm, find the corresponding altitude of the larger. 🎯", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "15 cm"}, "correct_answer": "B"},
{"id": 41, "difficulty": "hard", "question": "Two similar triangles have areas 81 cm² and 144 cm². If the perimeter of the smaller triangle is 36 cm, find the perimeter of the larger. 📏", "options": {"A": "42 cm", "B": "48 cm", "C": "54 cm", "D": "64 cm"}, "correct_answer": "B"},
{"id": 42, "difficulty": "hard", "question": "If ΔABC ~ ΔDEF and the ratio of their areas is 4:9, and BC + EF = 25 cm, find BC and EF individually. ➗", "options": {"A": "BC=10 cm, EF=15 cm", "B": "BC=12 cm, EF=13 cm", "C": "BC=8 cm, EF=17 cm", "D": "BC=15 cm, EF=10 cm"}, "correct_answer": "A"},
{"id": 43, "difficulty": "hard", "question": "The areas of two similar triangles are in the ratio 16:25. The sum of a pair of corresponding sides is 41 cm. Find these sides. 🔍", "options": {"A": "16 cm and 25 cm", "B": "20 cm and 21 cm", "C": "18 cm and 23 cm", "D": "24 cm and 17 cm"}, "correct_answer": "A"},
{"id": 44, "difficulty": "hard", "question": "Two similar triangles have their corresponding medians in the ratio 3:5. If the area of the smaller triangle is 54 cm², find the area of the larger. ⚕️", "options": {"A": "90 cm²", "B": "150 cm²", "C": "200 cm²", "D": "250 cm²"}, "correct_answer": "B"},
{"id": 45, "difficulty": "hard", "question": "If the ratio of the perimeters of two similar triangles is 5:8, and the area of the first is 100 cm², find the area of the second. 📐", "options": {"A": "160 cm²", "B": "200 cm²", "C": "256 cm²", "D": "320 cm²"}, "correct_answer": "C"},
{"id": 46, "difficulty": "hard", "question": "ΔPQR ~ ΔXYZ. Area(ΔPQR)=96 cm², QR=8 cm, YZ=12 cm. If PQ=10 cm, find XY. 🧮", "options": {"A": "12 cm", "B": "15 cm", "C": "18 cm", "D": "20 cm"}, "correct_answer": "B"},
{"id": 47, "difficulty": "hard", "question": "Two similar triangles have their corresponding altitudes in the ratio 4:7. If the area of the larger triangle is 245 cm², find the area of the smaller. 🏔️", "options": {"A": "64 cm²", "B": "80 cm²", "C": "100 cm²", "D": "120 cm²"}, "correct_answer": "B"},
{"id": 48, "difficulty": "hard", "question": "If the ratio of areas of two similar triangles is 36:49, and a side of the smaller triangle is 18 cm, find the length of the corresponding side of the larger triangle. 📏", "options": {"A": "21 cm", "B": "24 cm", "C": "28 cm", "D": "32 cm"}, "correct_answer": "A"},
{"id": 49, "difficulty": "hard", "question": "The sides of two similar triangles are in the ratio 2:3. If the area of the smaller triangle is 48 cm², and the perimeter of the larger triangle is 45 cm, find the perimeter of the smaller. 🔄", "options": {"A": "20 cm", "B": "30 cm", "C": "36 cm", "D": "40 cm"}, "correct_answer": "B"},
{"id": 50, "difficulty": "hard", "question": "Two similar triangles have their corresponding angle bisectors in the ratio 5:8. If the area of the smaller triangle is 75 cm², find the area of the larger. ⚡", "options": {"A": "120 cm²", "B": "192 cm²", "C": "200 cm²", "D": "240 cm²"}, "correct_answer": "B"},
{"id": 51, "difficulty": "hard", "question": "If ΔABC ~ ΔDEF, area(ΔABC)=121 cm², area(ΔDEF)=169 cm², and BC=11 cm, find EF. 🔺", "options": {"A": "12 cm", "B": "13 cm", "C": "14 cm", "D": "15 cm"}, "correct_answer": "B"},
{"id": 52, "difficulty": "hard", "question": "The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding perimeters. True or False? ✨", "options": {"A": "True", "B": "False", "C": "Only for equilateral", "D": "Only if scale factor <1"}, "correct_answer": "A"},
{"id": 53, "difficulty": "hard", "question": "Two similar triangles have areas 150 cm² and 216 cm². If a side of the first triangle is 25 cm, find the corresponding side of the second. 📐", "options": {"A": "30 cm", "B": "32 cm", "C": "36 cm", "D": "40 cm"}, "correct_answer": "A"},
{"id": 54, "difficulty": "hard", "question": "If the ratio of corresponding sides of two similar triangles is 7:10, and the difference of their areas is 204 cm², find the area of the larger triangle. ➖", "options": {"A": "300 cm²", "B": "400 cm²", "C": "500 cm²", "D": "600 cm²"}, "correct_answer": "B"},
{"id": 55, "difficulty": "hard", "question": "ΔLMN ~ ΔXYZ. Area(LMN)=64 cm², area(XYZ)=100 cm². If LM=8 cm, find XY. 🔗", "options": {"A": "10 cm", "B": "12 cm", "C": "14 cm", "D": "16 cm"}, "correct_answer": "A"},
{"id": 56, "difficulty": "hard", "question": "Two similar triangles have their corresponding medians in the ratio 4:9. If the area of the smaller triangle is 48 cm², find the area of the larger. ⚕️", "options": {"A": "108 cm²", "B": "162 cm²", "C": "243 cm²", "D": "324 cm²"}, "correct_answer": "C"},
{"id": 57, "difficulty": "hard", "question": "The perimeters of two similar triangles are 30 cm and 50 cm. If the area of the smaller is 45 cm², find the area of the larger. 📊", "options": {"A": "75 cm²", "B": "125 cm²", "C": "180 cm²", "D": "225 cm²"}, "correct_answer": "B"},
{"id": 58, "difficulty": "hard", "question": "If the ratio of the areas of two similar triangles is 1:4 and a side of the smaller triangle is 6 cm, find the length of the corresponding median in the larger triangle if the corresponding median in the smaller is 9 cm. 🧩", "options": {"A": "12 cm", "B": "15 cm", "C": "18 cm", "D": "21 cm"}, "correct_answer": "C"},
{"id": 59, "difficulty": "hard", "question": "Two similar triangles have their corresponding sides in the ratio 5:12. The sum of their areas is 338 cm². Find the area of the smaller triangle. ➕", "options": {"A": "50 cm²", "B": "72 cm²", "C": "98 cm²", "D": "128 cm²"}, "correct_answer": "A"},
{"id": 60, "difficulty": "hard", "question": "If ΔABC ~ ΔPQR, area(ΔABC)=225 cm², area(ΔPQR)=400 cm², and the altitude from A to BC is 15 cm, find the altitude from P to QR. 🏁", "options": {"A": "18 cm", "B": "20 cm", "C": "22 cm", "D": "24 cm"}, "correct_answer": "B"}
]
}