{
"title": "Tangent to Circle (Circles) - Grade 10 ICSE",
"total_questions": 60,
"questions": [
{"id": 1, "difficulty": "easy", "question": "A line which touches a circle at exactly one point is called a ______. 🔵", "options": {"A": "secant", "B": "chord", "C": "tangent", "D": "diameter"}, "correct_answer": "C"},
{"id": 2, "difficulty": "easy", "question": "At the point of contact, the tangent is ______ to the radius. 📐", "options": {"A": "parallel", "B": "perpendicular", "C": "equal", "D": "half"}, "correct_answer": "B"},
{"id": 3, "difficulty": "easy", "question": "How many tangents can be drawn from a point on the circle? 🎯", "options": {"A": "0", "B": "1", "C": "2", "D": "infinite"}, "correct_answer": "B"},
{"id": 4, "difficulty": "easy", "question": "If a line is perpendicular to the radius at its outer endpoint, it is ______ to the circle. 🔍", "options": {"A": "secant", "B": "tangent", "C": "chord", "D": "diameter"}, "correct_answer": "B"},
{"id": 5, "difficulty": "easy", "question": "Tangents drawn from an external point to a circle are ______ in length. ↔️", "options": {"A": "equal", "B": "unequal", "C": "half", "D": "double"}, "correct_answer": "A"},
{"id": 6, "difficulty": "easy", "question": "From point P outside circle, two tangents PA and PB. If PA = 7 cm, then PB = ______. 📏", "options": {"A": "3.5 cm", "B": "7 cm", "C": "14 cm", "D": "21 cm"}, "correct_answer": "B"},
{"id": 7, "difficulty": "easy", "question": "The length of tangent from a point inside the circle is ______. 🚫", "options": {"A": "zero", "B": "positive", "C": "negative", "D": "infinite"}, "correct_answer": "A"},
{"id": 8, "difficulty": "easy", "question": "Two tangents from an external point make ______ angles with the line joining the point to the center. ⚖️", "options": {"A": "acute", "B": "equal", "C": "supplementary", "D": "right"}, "correct_answer": "B"},
{"id": 9, "difficulty": "easy", "question": "If a line touches a circle, the perpendicular from the center to the line gives the ______. ⬇️", "options": {"A": "diameter", "B": "point of contact", "C": "external point", "D": "chord length"}, "correct_answer": "B"},
{"id": 10, "difficulty": "easy", "question": "A circle can have at most ______ parallel tangents. ↔️", "options": {"A": "1", "B": "2", "C": "3", "D": "infinite"}, "correct_answer": "B"},
{"id": 11, "difficulty": "easy", "question": "Tangents at the ends of a diameter are ______. 📏", "options": {"A": "perpendicular", "B": "parallel", "C": "equal", "D": "intersecting"}, "correct_answer": "B"},
{"id": 12, "difficulty": "easy", "question": "From point P, tangent PT touches circle at T. If OT = 5 cm and OP = 13 cm (O is center), then PT = ______. 🧮", "options": {"A": "8 cm", "B": "12 cm", "C": "15 cm", "D": "18 cm"}, "correct_answer": "B"},
{"id": 13, "difficulty": "easy", "question": "The angle between a tangent and a chord through the point of contact equals angle in the ______ segment. 🔄", "options": {"A": "same", "B": "opposite", "C": "alternate", "D": "major"}, "correct_answer": "C"},
{"id": 14, "difficulty": "easy", "question": "If a line is tangent to a circle, then the radius at the point of contact is ______ to it. 🔺", "options": {"A": "parallel", "B": "perpendicular", "C": "equal", "D": "half"}, "correct_answer": "B"},
{"id": 15, "difficulty": "easy", "question": "From an external point, the two tangents subtend ______ angles at the center. 🎯", "options": {"A": "equal", "B": "complementary", "C": "supplementary", "D": "right"}, "correct_answer": "A"},
{"id": 16, "difficulty": "easy", "question": "In the figure, if PQ is tangent and OQ is radius, then ∠OQP = ______. 📐", "options": {"A": "45°", "B": "60°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
{"id": 17, "difficulty": "easy", "question": "If two tangents from point P make an angle of 60°, the line joining P to center makes angle of ______ with each tangent. 🤔", "options": {"A": "15°", "B": "30°", "C": "60°", "D": "120°"}, "correct_answer": "B"},
{"id": 18, "difficulty": "easy", "question": "A tangent PQ touches circle at Q. Chord QA is drawn. ∠PQA is called the angle between tangent and chord. True or false? ✅", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
{"id": 19, "difficulty": "easy", "question": "The common point of a tangent and the circle is called ______. 📍", "options": {"A": "center", "B": "point of contact", "C": "external point", "D": "midpoint"}, "correct_answer": "B"},
{"id": 20, "difficulty": "easy", "question": "If a line touches two circles, it is called a ______ tangent. 🔗", "options": {"A": "single", "B": "common", "C": "internal", "D": "external"}, "correct_answer": "B"},
{"id": 21, "difficulty": "medium", "question": "From point P, two tangents PA & PB. If ∠APB = 80°, find ∠AOB. (O is center) 🧮", "options": {"A": "80°", "B": "100°", "C": "160°", "D": "180°"}, "correct_answer": "B"},
{"id": 22, "difficulty": "medium", "question": "Two circles of radii 4 cm and 9 cm touch externally. Length of direct common tangent = ? 📏", "options": {"A": "10 cm", "B": "12 cm", "C": "13 cm", "D": "15 cm"}, "correct_answer": "B"},
{"id": 23, "difficulty": "medium", "question": "Circle with center O, radius 8 cm. Tangent PQ touches at Q. If OQ = 8 cm and OP = 17 cm, find PQ. 📐", "options": {"A": "8 cm", "B": "15 cm", "C": "17 cm", "D": "25 cm"}, "correct_answer": "B"},
{"id": 24, "difficulty": "medium", "question": "If tangents at points A and B on a circle intersect at P, then which triangles are congruent? 🔺", "options": {"A": "△OAP & △OBP", "B": "△APB & △AOB", "C": "△OPA & △OPB", "D": "△PAB & △OAB"}, "correct_answer": "C"},
{"id": 25, "difficulty": "medium", "question": "Two concentric circles have radii 5 cm and 3 cm. Tangent from point on larger circle to smaller circle has length = ? 📏", "options": {"A": "2 cm", "B": "4 cm", "C": "6 cm", "D": "8 cm"}, "correct_answer": "B"},
{"id": 26, "difficulty": "medium", "question": "From point P, tangent PT and secant PAB drawn. If PT = 12 cm, PA = 8 cm, find AB. 🔄", "options": {"A": "4 cm", "B": "6 cm", "C": "8 cm", "D": "10 cm"}, "correct_answer": "D"},
{"id": 27, "difficulty": "medium", "question": "Two tangents TP & TQ from T. If ∠PTQ = 70°, find ∠OPQ. (O is center) 🔍", "options": {"A": "35°", "B": "55°", "C": "70°", "D": "110°"}, "correct_answer": "B"},
{"id": 28, "difficulty": "medium", "question": "In circle, tangent PQ touches at Q. Chord QN is drawn. If ∠PQN = 50°, find ∠QRN where R is point on major arc. 🔄", "options": {"A": "40°", "B": "50°", "C": "80°", "D": "100°"}, "correct_answer": "B"},
{"id": 29, "difficulty": "medium", "question": "Two circles touch externally. Their common tangents not through contact point are called ______. 🔗", "options": {"A": "transverse", "B": "direct", "C": "internal", "D": "external"}, "correct_answer": "B"},
{"id": 30, "difficulty": "medium", "question": "From point A, tangents AB & AC. If ∠BAC = 50°, find ∠BOC. (O is center) ⚖️", "options": {"A": "50°", "B": "100°", "C": "130°", "D": "180°"}, "correct_answer": "C"},
{"id": 31, "difficulty": "medium", "question": "In circle, tangent at point A. Chord AB is drawn. If ∠BAC = 40° where AC is another chord, find ∠ABC. 📐", "options": {"A": "40°", "B": "50°", "C": "80°", "D": "100°"}, "correct_answer": "A"},
{"id": 32, "difficulty": "medium", "question": "Two parallel tangents of a circle are at ends of a ______. ↔️", "options": {"A": "radius", "B": "diameter", "C": "chord", "D": "arc"}, "correct_answer": "B"},
{"id": 33, "difficulty": "medium", "question": "Circle with center O. Tangent PQ touches at Q. Chord QA is parallel to tangent at P. If ∠PQA = 35°, find ∠QPA. 🤔", "options": {"A": "35°", "B": "55°", "C": "70°", "D": "110°"}, "correct_answer": "C"},
{"id": 34, "difficulty": "medium", "question": "If from point P, two tangents PA & PB are inclined at 90°, then PAOB is a ______. (O is center) 🔲", "options": {"A": "square", "B": "rectangle", "C": "rhombus", "D": "kite"}, "correct_answer": "A"},
{"id": 35, "difficulty": "medium", "question": "Tangents PA & PB from P. If OP = 13 cm, radius = 5 cm, find perimeter of quadrilateral PAOB. 🔄", "options": {"A": "24 cm", "B": "30 cm", "C": "36 cm", "D": "42 cm"}, "correct_answer": "C"},
{"id": 36, "difficulty": "medium", "question": "Two circles intersect at A & B. From point P on AB produced, tangents PT & PS are equal because ______. ⚖️", "options": {"A": "P is external point", "B": "A,B are common points", "C": "tangents from same point", "D": "circles intersect"}, "correct_answer": "C"},
{"id": 37, "difficulty": "medium", "question": "In circle, tangent XY touches at Z. Chord ZA is drawn. If ∠XZA = 30°, find ∠AZB where B is point on circle. 🔄", "options": {"A": "30°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "A"},
{"id": 38, "difficulty": "medium", "question": "Two concentric circles. Chord of larger circle touching smaller circle is ______ of smaller circle. 🎯", "options": {"A": "tangent", "B": "secant", "C": "diameter", "D": "radius"}, "correct_answer": "A"},
{"id": 39, "difficulty": "medium", "question": "From point P, tangents PA & PB. If ∠APB = 60° and radius = 6 cm, find length of PA. 📏", "options": {"A": "6 cm", "B": "6√3 cm", "C": "12 cm", "D": "12√3 cm"}, "correct_answer": "B"},
{"id": 40, "difficulty": "medium", "question": "Circle inscribed in quadrilateral ABCD. If AB = 10 cm, BC = 7 cm, CD = 8 cm, find AD. 🔄", "options": {"A": "5 cm", "B": "6 cm", "C": "9 cm", "D": "11 cm"}, "correct_answer": "D"},
{"id": 41, "difficulty": "hard", "question": "Two circles of radii 5 cm and 3 cm have centers 10 cm apart. Length of transverse common tangent = ? 📏", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "B"},
{"id": 42, "difficulty": "hard", "question": "From point P, tangent PT and secant PAB drawn. If PA = 4 cm, AB = 5 cm, find PT. 🧮", "options": {"A": "3 cm", "B": "6 cm", "C": "9 cm", "D": "12 cm"}, "correct_answer": "B"},
{"id": 43, "difficulty": "hard", "question": "Two tangents TP & TQ from T. If ∠PTQ = 50°, find ∠OPQ. (O is center) 🔄", "options": {"A": "25°", "B": "40°", "C": "65°", "D": "130°"}, "correct_answer": "C"},
{"id": 44, "difficulty": "hard", "question": "Two circles touch externally at P. Common tangent not through P touches circles at A & B. If radii are 4 cm and 9 cm, find AB. 🔗", "options": {"A": "12 cm", "B": "13 cm", "C": "15 cm", "D": "17 cm"}, "correct_answer": "A"},
{"id": 45, "difficulty": "hard", "question": "In circle, tangent PQ touches at Q. Chord QR drawn parallel to tangent at P. If ∠PQR = 40°, find ∠QPR. 🔺", "options": {"A": "40°", "B": "50°", "C": "80°", "D": "100°"}, "correct_answer": "B"},
{"id": 46, "difficulty": "hard", "question": "Two circles intersect at A & B. From point P on AB produced, tangents PT & PS. If PT = 6 cm, PA = 4 cm, find PB. 📐", "options": {"A": "2 cm", "B": "3 cm", "C": "5 cm", "D": "9 cm"}, "correct_answer": "C"},
{"id": 47, "difficulty": "hard", "question": "Circle with center O. Tangent PT touches at T. Chord TA is perpendicular to radius OT. If ∠OTA = 30°, find ∠APT where P is external point. 🔍", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
{"id": 48, "difficulty": "hard", "question": "Two concentric circles. Chord of larger circle = 24 cm touches smaller circle of radius 5 cm. Find radius of larger circle. 📏", "options": {"A": "10 cm", "B": "12 cm", "C": "13 cm", "D": "15 cm"}, "correct_answer": "C"},
{"id": 49, "difficulty": "hard", "question": "Quadrilateral ABCD circumscribes circle touching at P,Q,R,S. If AB = 15 cm, BC = 12 cm, CD = 10 cm, find AD. 🔄", "options": {"A": "7 cm", "B": "9 cm", "C": "11 cm", "D": "13 cm"}, "correct_answer": "D"},
{"id": 50, "difficulty": "hard", "question": "Two tangents TP & TQ. If ∠PTQ = 80° and radius = 7 cm, find area of quadrilateral TPOQ. (O is center) 📦", "options": {"A": "49√3 cm²", "B": "98 cm²", "C": "98√3 cm²", "D": "196 cm²"}, "correct_answer": "C"},
{"id": 51, "difficulty": "hard", "question": "From point P, tangent PT = 12 cm, secant PAB with PA = 8 cm. A chord CD through P parallel to AB has PC = 3 cm. Find PD. 🧩", "options": {"A": "4 cm", "B": "4.5 cm", "C": "6 cm", "D": "9 cm"}, "correct_answer": "D"},
{"id": 52, "difficulty": "hard", "question": "Two circles of radii 8 cm and 3 cm have centers 13 cm apart. Length of direct common tangent = ? 📐", "options": {"A": "5 cm", "B": "12 cm", "C": "13 cm", "D": "15 cm"}, "correct_answer": "B"},
{"id": 53, "difficulty": "hard", "question": "Circle inscribed in quadrilateral ABCD touches sides at P,Q,R,S. If AP = 5 cm, BQ = 6 cm, CR = 4 cm, find DS. 📏", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "A"},
{"id": 54, "difficulty": "hard", "question": "Two circles touch externally. Their common tangents are perpendicular. Ratio of radii = ? ⚖️", "options": {"A": "1:2", "B": "1:3", "C": "1:4", "D": "1:5"}, "correct_answer": "B"},
{"id": 55, "difficulty": "hard", "question": "From point P, tangents PA & PB. Line through center O parallel to PA meets PB at Q. If PA = 10 cm, radius = 6 cm, find PQ. 📏", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "8 cm"}, "correct_answer": "B"},
{"id": 56, "difficulty": "hard", "question": "Two circles intersect at X & Y. Through X line AXB drawn. Through Y line CYD drawn. Tangents at A & C meet at P. Then A,P,C,Y are concyclic because of ______ theorem. 🔄", "options": {"A": "tangent-secant", "B": "alternate segment", "C": "intersecting chords", "D": "cyclic quadrilateral"}, "correct_answer": "B"},
{"id": 57, "difficulty": "hard", "question": "In circle, tangent PQ touches at Q. Chord QA parallel to tangent at P. If ∠PQA = 25°, find ∠QPA. 🔍", "options": {"A": "25°", "B": "50°", "C": "65°", "D": "75°"}, "correct_answer": "B"},
{"id": 58, "difficulty": "hard", "question": "Two circles of radii R and r touch externally. Length of direct common tangent is 2√Rr. True or false? ✅", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
{"id": 59, "difficulty": "hard", "question": "From point P, two tangents PA & PB and a line PCD cutting circle at C & D. If PC = 4 cm, CD = 5 cm, find PA using tangent-secant theorem. 🧮", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "6 cm"}, "correct_answer": "D"},
{"id": 60, "difficulty": "hard", "question": "Two tangents TP & TQ from T. If OP = 8 cm, PT = 15 cm (O center), find area of triangle OPT. 🔺", "options": {"A": "30 cm²", "B": "45 cm²", "C": "60 cm²", "D": "75 cm²"}, "correct_answer": "C"}
]
}