{
  "title": "Theorems on Areas (Area Theorems) - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Triangles on same base and between same parallels are ______ in area. 📐", "options": {"A": "different", "B": "equal", "C": "double", "D": "half"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "A median of triangle divides it into two triangles of ______ area.", "options": {"A": "different", "B": "equal", "C": "triple", "D": "quarter"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "If a triangle and parallelogram are on same base & between parallels, area triangle = ______ parallelogram.", "options": {"A": "same", "B": "half", "C": "double", "D": "one third"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "Two triangles with equal bases and equal heights have ______ area.", "options": {"A": "unequal", "B": "equal", "C": "proportional", "D": "zero"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "Parallelograms on same base & between parallels are ______ in area.", "options": {"A": "different", "B": "equal", "C": "triple", "D": "half"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If two triangles have same base and equal areas, their vertices lie on line ______ to base.", "options": {"A": "perpendicular", "B": "parallel", "C": "coincident", "D": "intersecting"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "Area of triangle = ½ × base × ______.", "options": {"A": "side", "B": "height", "C": "perimeter", "D": "diagonal"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "Diagonals of parallelogram divide it into ______ triangles of equal area.", "options": {"A": "two", "B": "four", "C": "three", "D": "six"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Two triangles between same parallels have equal ______.", "options": {"A": "bases", "B": "heights", "C": "areas", "D": "perimeters"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "A line through midpoint of one side of triangle parallel to another side bisects ______ side.", "options": {"A": "first", "B": "second", "C": "third", "D": "none"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "If two parallelograms are equal in area and on same base, they are between ______ parallels.", "options": {"A": "different", "B": "same", "C": "intersecting", "D": "perpendicular"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "The area of triangle is ______ area of parallelogram on same base & height.", "options": {"A": "equal to", "B": "half of", "C": "double of", "D": "one-third of"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "A diagonal divides parallelogram into two ______ triangles.", "options": {"A": "congruent", "B": "similar", "C": "different", "D": "right"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "In two triangles with equal bases, ratio of areas = ratio of ______.", "options": {"A": "heights", "B": "angles", "C": "perimeters", "D": "medians"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "If heights are same, area is proportional to ______.", "options": {"A": "base", "B": "angles", "C": "sides", "D": "diagonals"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "To have equal areas with same base, triangles must have same ______.", "options": {"A": "height", "B": "angles", "C": "vertices", "D": "perimeter"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "If triangle area = parallelogram area on same base, then height ratio (triangle:parallelogram) = ?", "options": {"A": "1:1", "B": "2:1", "C": "1:2", "D": "3:1"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "Midpoint theorem relates to ______ of triangle.", "options": {"A": "area", "B": "parallel sides", "C": "line joining midpoints", "D": "angles"}, "correct_answer": "C"},
    {"id": 19, "difficulty": "easy", "question": "Between same parallels means ______ distance is constant.", "options": {"A": "vertical", "B": "horizontal", "C": "perpendicular", "D": "diagonal"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "In trapezium, triangles on same base formed by diagonals are ______ in area.", "options": {"A": "equal", "B": "different", "C": "double", "D": "half"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "Given: parallelogram ABCD, diagonal AC. Area(ABC) = area(ADC). This is because ______.", "options": {"A": "same base & height", "B": "congruent triangles", "C": "both", "D": "midpoint theorem"}, "correct_answer": "C"},
    {"id": 22, "difficulty": "medium", "question": "Two triangles ABC and DBC have same base BC. If area(ABC)=area(DBC), then AD must be ______ to BC.", "options": {"A": "equal", "B": "parallel", "C": "perpendicular", "D": "coincident"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "Triangle PQR, S midpoint of QR. Area(PQS) : Area(PQR) = ?", "options": {"A": "1:1", "B": "1:2", "C": "1:3", "D": "2:1"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "In quadrilateral ABCD, diagonal AC divides it into two triangles. They have equal area only if AC is ______.", "options": {"A": "bisector", "B": "not necessarily bisected", "C": "perpendicular", "D": "parallel"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "A line through vertex of triangle divides it into two equal areas. That line is a ______.", "options": {"A": "median", "B": "altitude", "C": "angle bisector", "D": "perpendicular"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "If two triangles have equal areas and equal bases, they must have equal ______.", "options": {"A": "heights", "B": "angles", "C": "perimeters", "D": "medians"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "Parallelogram and rectangle on same base & between parallels have ______ area.", "options": {"A": "different", "B": "equal", "C": "double", "D": "half"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "In triangle, three medians divide it into ______ triangles of equal area.", "options": {"A": "2", "B": "3", "C": "6", "D": "4"}, "correct_answer": "C"},
    {"id": 29, "difficulty": "medium", "question": "If two triangles stand on equal bases and between same parallels, their areas are ______.", "options": {"A": "proportional", "B": "equal", "C": "different", "D": "half"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "Two triangles with same height have areas proportional to their ______.", "options": {"A": "bases", "B": "angles", "C": "sides", "D": "perimeters"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "In trapezium ABCD with AB∥DC, area(ABD) = area(ABC) because ______.", "options": {"A": "same base AB, same height", "B": "congruent", "C": "parallel sides", "D": "angles equal"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "If triangle area doubles with same base, height must ______.", "options": {"A": "double", "B": "halve", "C": "same", "D": "triple"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "A line divides two sides of triangle in same ratio. It divides area in ______ ratio.", "options": {"A": "same", "B": "square of", "C": "cube of", "D": "sum of"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "Two triangles between same parallels and on equal bases are ______.", "options": {"A": "congruent", "B": "similar", "C": "equal area", "D": "different shape"}, "correct_answer": "C"},
    {"id": 35, "difficulty": "medium", "question": "Triangle area = ½bh. If b constant, area proportional to ______.", "options": {"A": "h", "B": "b²", "C": "h²", "D": "b+h"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "If triangle and parallelogram have equal areas on same base, height of triangle = ______ height of parallelogram.", "options": {"A": "same", "B": "double", "C": "half", "D": "triple"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "A median divides triangle into two equal areas because they have same ______.", "options": {"A": "base and height", "B": "angles", "C": "sides", "D": "perimeter"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "In two triangles with equal bases, if area1:area2 = 3:2, then height1:height2 = ?", "options": {"A": "3:2", "B": "2:3", "C": "9:4", "D": "1:1"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Parallelogram area = base × height. Triangle area = ______ × base × height.", "options": {"A": "1", "B": "½", "C": "2", "D": "¼"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "Triangles with equal heights: area ratio = ______ ratio.", "options": {"A": "base", "B": "angle", "C": "side", "D": "perimeter"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "In ∆ABC, D,E,F midpoints of BC,CA,AB. Area(DEF) = ? × Area(ABC).", "options": {"A": "¼", "B": "½", "C": "⅓", "D": "⅛"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "If through midpoint of median, line ∥ base is drawn, area of small triangle = ______ original.", "options": {"A": "⅛", "B": "¼", "C": "½", "D": "⅓"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "Two triangles on same base and equal areas have vertices on line ______ to base.", "options": {"A": "parallel", "B": "perpendicular", "C": "coincident", "D": "same"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "In quadrilateral, diagonal divides into triangles. They equal area if diagonal is ______.", "options": {"A": "bisected", "B": "not necessary", "C": "perpendicular", "D": "parallel"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "hard", "question": "If triangle area = 48 cm², base = 12 cm, height from same vertex to extended base if line ∥ base through midpoint of side? 🤔", "options": {"A": "4 cm", "B": "8 cm", "C": "12 cm", "D": "6 cm"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "hard", "question": "Triangle PQR, S on PQ such that PS:SQ=2:1, T on PR such that PT:TR=2:1. Area(PST) : Area(PQR) = ?", "options": {"A": "4:9", "B": "2:3", "C": "1:3", "D": "2:9"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "Given parallelogram ABCD, point E on BC. Area(ABE) + area(ECD) = ______ area(ABCD).", "options": {"A": "½", "B": "¼", "C": "¾", "D": "equal"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "In trapezium ABCD, AB∥DC. Diagonals AC & BD intersect at O. Area(AOD) = area(BOC) because ______.", "options": {"A": "same base & height", "B": "both subtractions from equal triangles", "C": "congruent", "D": "similar"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "hard", "question": "Two triangles ABC, DEF have equal areas. AB=DE, and heights equal. Then BC = ______.", "options": {"A": "EF", "B": "different", "C": "twice", "D": "half"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "In ∆ABC, D on BC such that BD:DC=3:2. Area(ABD) : Area(ADC) = ?", "options": {"A": "3:2", "B": "2:3", "C": "9:4", "D": "4:9"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "If a triangle’s base increased by 30% and height decreased by 30%, area change % = ?", "options": {"A": "-9%", "B": "+9%", "C": "-30%", "D": "0%"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "Triangle ABC, D,E,F midpoints. Area of triangle formed by joining midpoints of sides of DEF = ? × area(ABC).", "options": {"A": "1/16", "B": "1/8", "C": "1/4", "D": "1/2"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "Two triangles with same base have areas in ratio 4:5. Their heights ratio?", "options": {"A": "4:5", "B": "5:4", "C": "16:25", "D": "2:√5"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "In parallelogram ABCD, P any point on AB. Area(APD) + area(BPC) = ______ area(ABCD).", "options": {"A": "½", "B": "¼", "C": "¾", "D": "1"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Given triangle area 72 cm², base 18 cm. Find height of triangle formed by joining midpoints of sides.", "options": {"A": "4 cm", "B": "6 cm", "C": "8 cm", "D": "12 cm"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "hard", "question": "Triangles on equal bases and between same parallels have ______ area.", "options": {"A": "equal", "B": "different", "C": "proportional", "D": "half"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "ABCD is a parallelogram. E is midpoint of AB. Area(ADE) = ? × area(ABCD).", "options": {"A": "¼", "B": "½", "C": "⅛", "D": "⅓"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "If triangle area formula is ½bh, for parallelogram on same base & height, area = ______.", "options": {"A": "bh", "B": "½bh", "C": "2bh", "D": "b+h"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "Through midpoint of median of triangle, line ∥ base divides triangle into areas ratio ______.", "options": {"A": "1:3", "B": "1:7", "C": "1:8", "D": "1:4"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "hard", "question": "In ∆ABC, AD median, E midpoint of AD. BE extended meets AC at F. Area(BFD) : Area(ABC) = ?", "options": {"A": "1:4", "B": "1:6", "C": "1:8", "D": "1:12"}, "correct_answer": "B"}
  ]
}