{
  "title": "Proofs (Area Theorems) - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "To prove two triangles have equal area, they must have same base and ______. 📏", "options": {"A": "angles", "B": "height", "C": "perimeter", "D": "vertices"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "If two triangles are between same parallels, their ______ are equal.", "options": {"A": "bases", "B": "heights", "C": "areas", "D": "sides"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "A median divides a triangle into two triangles of ______ area.", "options": {"A": "different", "B": "double", "C": "equal", "D": "half"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "Two triangles on same base and between same parallels are ______ in area.", "options": {"A": "different", "B": "equal", "C": "triple", "D": "half"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "Area of triangle = ½ × base × ______. 📐", "options": {"A": "side", "B": "height", "C": "diagonal", "D": "angle"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If a triangle and parallelogram are on same base and between parallels, area of triangle = ______ of parallelogram.", "options": {"A": "same", "B": "double", "C": "half", "D": "one-third"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "In proofs, we often use ______ to show two shapes have same height.", "options": {"A": "parallel lines", "B": "circles", "C": "angles", "D": "curves"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "Two triangles with equal bases and equal heights have ______ areas.", "options": {"A": "proportional", "B": "equal", "C": "different", "D": "zero"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "If heights are equal and bases equal, then areas are ______.", "options": {"A": "unequal", "B": "equal", "C": "double", "D": "half"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "A parallelogram divided by a diagonal gives two triangles of ______ area. ⚖️", "options": {"A": "different", "B": "equal", "C": "triple", "D": "quarter"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "To prove area theorem, we compare triangle with ______.", "options": {"A": "circle", "B": "rectangle", "C": "parallelogram", "D": "hexagon"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "If two triangles have same base and lie between same parallels, their heights are ______.", "options": {"A": "different", "B": "equal", "C": "zero", "D": "infinite"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "Triangle area is ______ area of rectangle on same base & height.", "options": {"A": "equal to", "B": "half of", "C": "double of", "D": "same as"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "A line joining mid-points of two sides divides triangle into two ______ areas.", "options": {"A": "equal", "B": "unequal", "C": "double", "D": "triple"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "In area theorem proofs, congruent triangles have ______ area.", "options": {"A": "different", "B": "equal", "C": "no", "D": "triple"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "If triangles have equal area and same base, they must be between same ______.", "options": {"A": "angles", "B": "parallels", "C": "vertices", "D": "circles"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "Two triangles with equal area and same height have ______ bases.", "options": {"A": "equal", "B": "proportional", "C": "different", "D": "zero"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "A diagonal of parallelogram divides it into two ______ triangles.", "options": {"A": "congruent", "B": "different", "C": "right", "D": "scalene"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Proof of area theorem uses concept of ______ figures.", "options": {"A": "3D", "B": "similar", "C": "equal", "D": "parallel"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "If two triangles have equal area, their base-height product is ______.", "options": {"A": "different", "B": "same", "C": "zero", "D": "double"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "Given: parallelogram ABCD, diagonal AC. Prove area(ABC) = area(ADC). Which theorem?", "options": {"A": "Midpoint theorem", "B": "Parallelogram diagonal divides equally", "C": "Pythagoras", "D": "Area of rectangle"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "Two triangles between same parallels have ______.", "options": {"A": "same bases", "B": "same heights", "C": "same perimeters", "D": "same sides"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "If a triangle and parallelogram are on same base & between parallels, ratio of areas = ______.", "options": {"A": "1:1", "B": "1:2", "C": "2:1", "D": "3:1"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "In ∆ABC, AD is median. Area(ABD) : Area(ADC) = ?", "options": {"A": "1:1", "B": "1:2", "C": "2:1", "D": "1:3"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "To prove area(∆PQR) = area(∆SQR), what must be true?", "options": {"A": "Same height from P & S to QR", "B": "PQ = SR", "C": "Angle P = Angle S", "D": "PR = QS"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "If two triangles have equal areas and common base, their vertices lie on a line ______ to base.", "options": {"A": "perpendicular", "B": "parallel", "C": "inclined", "D": "tangent"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "Triangle PQR and triangle QRS share base QR. If areas equal, then P and S are on line parallel to ______.", "options": {"A": "PQ", "B": "QR", "C": "RS", "D": "PS"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "A quadrilateral’s diagonal divides it into two triangles of ______ area if diagonal is ______.", "options": {"A": "equal, median", "B": "equal, bisector", "C": "different, same", "D": "equal, not always equal"}, "correct_answer": "D"},
    {"id": 29, "difficulty": "medium", "question": "In theorem: Triangles on same base & between same parallels have ______ area.", "options": {"A": "different", "B": "equal", "C": "triple", "D": "half"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "To prove two triangles have equal area without height, we check they are between ______.", "options": {"A": "same angles", "B": "same parallels", "C": "same circles", "D": "same vertices"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "If ∆ABC and ∆DBC have same base BC, and area(ABC)=area(DBC), then AD must be ______ to BC.", "options": {"A": "equal", "B": "parallel", "C": "perpendicular", "D": "touching"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "Proof that median divides triangle into equal areas uses concept of ______.", "options": {"A": "same base & height", "B": "similar triangles", "C": "Pythagoras", "D": "circle theorem"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "In ∆XYZ, A and B are midpoints of XY and XZ. Area(XAB) = ? part of area(XYZ).", "options": {"A": "1/2", "B": "1/3", "C": "1/4", "D": "1/8"}, "correct_answer": "C"},
    {"id": 34, "difficulty": "medium", "question": "Two parallelograms on same base & between parallels have ______ area.", "options": {"A": "different", "B": "equal", "C": "double", "D": "triple"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "If a triangle’s base is doubled & height halved, new area = ______ old area.", "options": {"A": "same as", "B": "double", "C": "half", "D": "quarter"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "To prove area(∆ABD)=area(∆ACD) when D is midpoint of BC, we say they have same ______.", "options": {"A": "base", "B": "height", "C": "both A & B", "D": "angles"}, "correct_answer": "C"},
    {"id": 37, "difficulty": "medium", "question": "In theorem proofs, ‘between same parallels’ means ______ distance between lines is constant.", "options": {"A": "vertical", "B": "horizontal", "C": "shortest", "D": "diagonal"}, "correct_answer": "C"},
    {"id": 38, "difficulty": "medium", "question": "Triangle and rectangle on same base & height: area ratio = ______.", "options": {"A": "1:2", "B": "1:1", "C": "2:1", "D": "3:2"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "If triangles have same area and equal bases, then their heights are ______.", "options": {"A": "equal", "B": "different", "C": "zero", "D": "infinite"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "Proof of ‘triangles on same base…’ uses construction of drawing ______ line.", "options": {"A": "parallel", "B": "perpendicular", "C": "tangent", "D": "dashed"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "Given: ∆ABC, D,E midpoints of AB, AC. Prove area(ADE)=¼ area(ABC). Which theorem is used? 🤔", "options": {"A": "Midpoint theorem", "B": "Basic proportionality", "C": "Area = ½ bh", "D": "Both A & C"}, "correct_answer": "D"},
    {"id": 42, "difficulty": "hard", "question": "Two triangles ABC and DBC have same base BC. Area(ABC)=area(DBC). Prove AD ∥ BC.", "options": {"A": "By height equality", "B": "By midpoint theorem", "C": "By congruency", "D": "By Pythagoras"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "If a line divides two sides of triangle proportionally, it divides area in ______ ratio.", "options": {"A": "same", "B": "square of", "C": "cube of", "D": "inverse"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "hard", "question": "In ∆PQR, S is point on QR such that QS:SR=2:1. Area(PQS):Area(PSR)=?", "options": {"A": "2:1", "B": "1:2", "C": "4:1", "D": "1:1"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "hard", "question": "Prove: Triangles between same parallels & on equal bases are equal in area. Uses ______.", "options": {"A": "Congruency", "B": "Area formula", "C": "Both", "D": "Similarity"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "hard", "question": "ABCD parallelogram, E any point on BC. Area(ABE) + area(ECD) = ______ area(ABCD).", "options": {"A": "½", "B": "¼", "C": "¾", "D": "equal"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "If two triangles have equal area and stand on same base, vertices are on line ______ to base.", "options": {"A": "perpendicular", "B": "parallel", "C": "coincident", "D": "intersecting"}, "correct_answer": "B"},
    {"id": 48, "difficulty": "hard", "question": "In quadrilateral ABCD, diagonal AC not bisected. Area(ABC) ______ area(ADC).", "options": {"A": "equals", "B": "not necessarily equals", "C": "double", "D": "half"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "hard", "question": "Triangle area ratio for two triangles with same height is ______ ratio of bases.", "options": {"A": "same as", "B": "square of", "C": "inverse of", "D": "sum of"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "Prove: A median divides triangle into equal areas. Which property is key?", "options": {"A": "Same base & height", "B": "Congruent parts", "C": "Parallel lines", "D": "Right angles"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "If triangles share vertex and base on same line, with equal area, then bases are ______.", "options": {"A": "equal", "B": "proportional to heights", "C": "different", "D": "parallel"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "hard", "question": "In ∆ABC, D,E,F midpoints. Area(DEF) = ? × area(ABC).", "options": {"A": "½", "B": "¼", "C": "⅛", "D": "¾"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "hard", "question": "Given parallelogram & triangle on same base between parallels, proof uses ______ to show area relation.", "options": {"A": "diagonal construction", "B": "perpendicular drop", "C": "midline", "D": "angle bisector"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "Two triangles with equal area and common base have vertices on line ______ to base.", "options": {"A": "parallel", "B": "perpendicular", "C": "same", "D": "opposite"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Proof: Triangles on equal bases and between same parallels are equal in area. Which step first?", "options": {"A": "Draw heights", "B": "Join vertices", "C": "Construct parallelogram", "D": "Use Pythagoras"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "In theorem proving equal areas, if bases equal and heights equal, then areas ______.", "options": {"A": "equal", "B": "different", "C": "double", "D": "half"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "If ∆ABC and ∆DEF have equal areas and equal bases, then their heights are ______.", "options": {"A": "equal", "B": "unequal", "C": "zero", "D": "infinite"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "ABCD trapezium, AB∥DC. Diagonals divide it into triangles. Which pair equal area?", "options": {"A": "Area(ABD)=Area(ABC)", "B": "Area(ADC)=Area(BCD)", "C": "Area(ABD)=Area(ACD)", "D": "None"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "hard", "question": "If through midpoint of median, line ∥ base is drawn, area of small triangle = ______ original.", "options": {"A": "¼", "B": "⅛", "C": "½", "D": "⅓"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "hard", "question": "Proving two triangles equal in area without numbers uses ______ principles.", "options": {"A": "geometric", "B": "algebraic", "C": "trigonometric", "D": "calculus"}, "correct_answer": "A"}
  ]
}