{
"title": "Properties of Tangents (Circles) - Grade 10 ICSE",
"total_questions": 60,
"questions": [
{"id": 1, "difficulty": "easy", "question": "A tangent to a circle touches it at exactly ______ point(s). 🔴", "options": {"A": "0", "B": "1", "C": "2", "D": "infinite"}, "correct_answer": "B"},
{"id": 2, "difficulty": "easy", "question": "The angle between a tangent and the radius at the point of contact is ______. 📐", "options": {"A": "acute", "B": "obtuse", "C": "right", "D": "straight"}, "correct_answer": "C"},
{"id": 3, "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": 4, "difficulty": "easy", "question": "From point P outside circle, two tangents PA and PB are drawn. If PA = 5 cm, then PB = ______. 📏", "options": {"A": "2.5 cm", "B": "5 cm", "C": "10 cm", "D": "15 cm"}, "correct_answer": "B"},
{"id": 5, "difficulty": "easy", "question": "The line joining the external point to the center of the circle ______ the angle between the tangents. ✂️", "options": {"A": "bisects", "B": "doubles", "C": "triples", "D": "equals"}, "correct_answer": "A"},
{"id": 6, "difficulty": "easy", "question": "How many tangents can be drawn from a point inside the circle? 🚫", "options": {"A": "0", "B": "1", "C": "2", "D": "infinite"}, "correct_answer": "A"},
{"id": 7, "difficulty": "easy", "question": "If a line touches a circle, it is called a ______. 📍", "options": {"A": "secant", "B": "chord", "C": "tangent", "D": "diameter"}, "correct_answer": "C"},
{"id": 8, "difficulty": "easy", "question": "The point where a tangent touches the circle is called the ______. 🎯", "options": {"A": "center", "B": "point of contact", "C": "external point", "D": "midpoint"}, "correct_answer": "B"},
{"id": 9, "difficulty": "easy", "question": "From an external point, the maximum number of tangents to a circle is ______. 🔢", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
{"id": 10, "difficulty": "easy", "question": "If radius OT is perpendicular to tangent PQ at T, then ∠OTQ = ______. 🧮", "options": {"A": "45°", "B": "60°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
{"id": 11, "difficulty": "easy", "question": "Two tangents TP and TQ from point T to circle with center O. If ∠PTQ = 60°, then ∠POQ = ______. ⚖️", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "180°"}, "correct_answer": "C"},
{"id": 12, "difficulty": "easy", "question": "In the figure with two tangents from point P, if ∠APB = 70°, then ∠AOB = ______. (O is center) 📐", "options": {"A": "70°", "B": "110°", "C": "140°", "D": "180°"}, "correct_answer": "B"},
{"id": 13, "difficulty": "easy", "question": "The lengths of tangents from an external point are equal. True or false? ✅", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
{"id": 14, "difficulty": "easy", "question": "A tangent is perpendicular to the radius through the point of contact. This is a ______. 📚", "options": {"A": "theorem", "B": "axiom", "C": "definition", "D": "postulate"}, "correct_answer": "A"},
{"id": 15, "difficulty": "easy", "question": "A line intersecting a circle at two points is called a ______. 🔪", "options": {"A": "tangent", "B": "secant", "C": "diameter", "D": "radius"}, "correct_answer": "B"},
{"id": 16, "difficulty": "easy", "question": "If two tangents are inclined at 90°, the line joining external point to center makes angle of ______ with each tangent. 🤔", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
{"id": 17, "difficulty": "easy", "question": "In a circle, the perpendicular from center to a tangent gives ______. ⬇️", "options": {"A": "radius", "B": "diameter", "C": "point of contact", "D": "external point"}, "correct_answer": "C"},
{"id": 18, "difficulty": "easy", "question": "From point P, tangents PA and PB. If OA = 3 cm and OP = 5 cm (O is center), then PA = ______. 📏", "options": {"A": "3 cm", "B": "4 cm", "C": "5 cm", "D": "8 cm"}, "correct_answer": "B"},
{"id": 19, "difficulty": "easy", "question": "A tangent PQ touches circle at Q. O is center. Which triangle is right-angled? 🔺", "options": {"A": "△OPQ", "B": "△OQP", "C": "△POQ", "D": "All"}, "correct_answer": "A"},
{"id": 20, "difficulty": "easy", "question": "Tangents at ends of a diameter are ______. ↔️", "options": {"A": "parallel", "B": "perpendicular", "C": "equal", "D": "intersecting"}, "correct_answer": "A"},
{"id": 21, "difficulty": "medium", "question": "From point P, two tangents PA & PB. If ∠APB = 50°, find ∠AOB. (O is center) 🧮", "options": {"A": "50°", "B": "100°", "C": "130°", "D": "180°"}, "correct_answer": "C"},
{"id": 22, "difficulty": "medium", "question": "Two concentric circles have radii 5 cm and 3 cm. Length of tangent from point on larger circle to smaller circle is ______. 📏", "options": {"A": "2 cm", "B": "4 cm", "C": "6 cm", "D": "8 cm"}, "correct_answer": "B"},
{"id": 23, "difficulty": "medium", "question": "Tangents PA & PB from point P to circle with center O. If OP = 10 cm, radius = 6 cm, find perimeter of quadrilateral PAOB. 🔄", "options": {"A": "20 cm", "B": "24 cm", "C": "28 cm", "D": "32 cm"}, "correct_answer": "C"},
{"id": 24, "difficulty": "medium", "question": "Two circles touch externally. Common tangents not passing through point of contact are called ______. 🔗", "options": {"A": "direct common tangents", "B": "transverse common tangents", "C": "internal tangents", "D": "external tangents"}, "correct_answer": "A"},
{"id": 25, "difficulty": "medium", "question": "Circle with center O, radius 5 cm. From point P, tangent PT of length 12 cm. Find OP. 📐", "options": {"A": "7 cm", "B": "13 cm", "C": "17 cm", "D": "25 cm"}, "correct_answer": "B"},
{"id": 26, "difficulty": "medium", "question": "In 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": 27, "difficulty": "medium", "question": "If ∠OAP = 90° where AP is tangent and O is center, which side is hypotenuse in right triangle? 🔺", "options": {"A": "OA", "B": "AP", "C": "OP", "D": "OA or AP"}, "correct_answer": "C"},
{"id": 28, "difficulty": "medium", "question": "From point P, tangents PA & PB. If PA = 8 cm and radius = 6 cm, find OP. 📏", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "14 cm"}, "correct_answer": "B"},
{"id": 29, "difficulty": "medium", "question": "Two tangents TP & TQ. If ∠PTQ = 70°, find ∠OPQ. (O is center) 🤔", "options": {"A": "35°", "B": "55°", "C": "70°", "D": "110°"}, "correct_answer": "B"},
{"id": 30, "difficulty": "medium", "question": "Circle with diameter AB. Tangent at A and line through B meets at P. If ∠P = 40°, find ∠ABP. 📐", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "80°"}, "correct_answer": "B"},
{"id": 31, "difficulty": "medium", "question": "Two circles of radii 4 cm and 9 cm touch externally. Length of direct common tangent = ? 📏", "options": {"A": "12 cm", "B": "13 cm", "C": "14 cm", "D": "15 cm"}, "correct_answer": "A"},
{"id": 32, "difficulty": "medium", "question": "If two tangents are parallel, they touch circle at ends of a ______. ↔️", "options": {"A": "radius", "B": "diameter", "C": "chord", "D": "arc"}, "correct_answer": "B"},
{"id": 33, "difficulty": "medium", "question": "Tangents PA & PB from P. If ∠APB = 60° and PA = 10 cm, find length of chord AB. 🧵", "options": {"A": "5 cm", "B": "10 cm", "C": "15 cm", "D": "20 cm"}, "correct_answer": "B"},
{"id": 34, "difficulty": "medium", "question": "From point P, tangent PT and secant PAB are drawn. If PT = 6 cm, PA = 4 cm, find AB. 🔄", "options": {"A": "2 cm", "B": "3 cm", "C": "5 cm", "D": "9 cm"}, "correct_answer": "C"},
{"id": 35, "difficulty": "medium", "question": "Two circles intersect at A & B. From point P on AB produced, tangents PT & PS are drawn. Then PT ______ PS. ⚖️", "options": {"A": ">", "B": "<", "C": "=", "D": "≠"}, "correct_answer": "C"},
{"id": 36, "difficulty": "medium", "question": "If tangents at points A & B on circle intersect at P, then PA = PB and ∠APO = ∠BPO (O is center). This makes triangles ______. 🔺", "options": {"A": "similar", "B": "congruent", "C": "isosceles", "D": "right"}, "correct_answer": "B"},
{"id": 37, "difficulty": "medium", "question": "In circle, tangent PQ touches at Q. Chord QA is drawn. ∠PQA = 40°. Find ∠AQB if QB is diameter. 📐", "options": {"A": "40°", "B": "50°", "C": "80°", "D": "100°"}, "correct_answer": "B"},
{"id": 38, "difficulty": "medium", "question": "Two concentric circles. Chord of larger circle touching smaller circle is bisected at point of contact. True or false? ✅", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "A"},
{"id": 39, "difficulty": "medium", "question": "If from point P, two tangents PA & PB are inclined at 70°, find ∠OAP. (O is center) 🤓", "options": {"A": "20°", "B": "35°", "C": "55°", "D": "70°"}, "correct_answer": "C"},
{"id": 40, "difficulty": "medium", "question": "Circle with center O, radius 8 cm. From point P, tangent PT = 15 cm. Find area of quadrilateral OTPA if PA is another tangent? 📦", "options": {"A": "120 cm²", "B": "136 cm²", "C": "150 cm²", "D": "164 cm²"}, "correct_answer": "A"},
{"id": 41, "difficulty": "hard", "question": "Two circles of radii 5 cm and 3 cm have their 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 outside circle, tangent PT and secant PAB drawn. If PA = 9 cm, AB = 7 cm, find PT. 🧮", "options": {"A": "12 cm", "B": "14 cm", "C": "16 cm", "D": "18 cm"}, "correct_answer": "A"},
{"id": 43, "difficulty": "hard", "question": "Two tangents TP & TQ from T to circle with center O. ∠PTQ = 50°. Find ∠OPQ. 🔄", "options": {"A": "25°", "B": "40°", "C": "50°", "D": "65°"}, "correct_answer": "D"},
{"id": 44, "difficulty": "hard", "question": "Circle with center O, radius 6 cm. Quadrilateral ABCD circumscribes circle touching at P,Q,R,S. If AB = 10 cm, CD = 8 cm, find BC+AD. 🔄", "options": {"A": "12 cm", "B": "14 cm", "C": "16 cm", "D": "18 cm"}, "correct_answer": "D"},
{"id": 45, "difficulty": "hard", "question": "Two circles touch internally at P. Chord PQ of larger circle touches smaller circle at R. Prove PR = RQ uses which theorem? 🧠", "options": {"A": "Tangent-secant theorem", "B": "Alternate segment theorem", "C": "Pythagoras theorem", "D": "Intersecting chords"}, "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 = 8 cm, PA = 6 cm, find PB. 📐", "options": {"A": "4 cm", "B": "5 cm", "C": "6 cm", "D": "7 cm"}, "correct_answer": "B"},
{"id": 47, "difficulty": "hard", "question": "In circle, tangent PQ touches at Q. Chord QR is drawn parallel to tangent at P. If ∠PQR = 35°, find ∠QPR. 🔺", "options": {"A": "35°", "B": "55°", "C": "70°", "D": "110°"}, "correct_answer": "B"},
{"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": "From point A outside circle, tangents AB & AC. Line through center O meets chord BC at D. If OD = 3 cm, radius = 5 cm, find BC. 📐", "options": {"A": "6 cm", "B": "8 cm", "C": "10 cm", "D": "12 cm"}, "correct_answer": "B"},
{"id": 50, "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": 51, "difficulty": "hard", "question": "Circle inscribed in quadrilateral ABCD touches sides at P,Q,R,S. If AB = 12 cm, BC = 14 cm, CD = 10 cm, find AD. 📏", "options": {"A": "8 cm", "B": "10 cm", "C": "12 cm", "D": "14 cm"}, "correct_answer": "A"},
{"id": 52, "difficulty": "hard", "question": "Two tangents TP & TQ. If ∠PTQ = 80° and radius = 5 cm, find area of quadrilateral TPOQ. (O is center) 📦", "options": {"A": "25√3 cm²", "B": "50 cm²", "C": "50√3 cm²", "D": "100 cm²"}, "correct_answer": "C"},
{"id": 53, "difficulty": "hard", "question": "From point P, tangent PT = 10 cm, secant PAB with PA = 6 cm. A chord CD through P parallel to AB has PC = 4 cm. Find PD. 🧩", "options": {"A": "5 cm", "B": "6 cm", "C": "7.5 cm", "D": "9 cm"}, "correct_answer": "D"},
{"id": 54, "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 ______. 🔄", "options": {"A": "tangent properties", "B": "alternate segment theorem", "C": "opposite angles supplementary", "D": "equal tangents"}, "correct_answer": "B"},
{"id": 55, "difficulty": "hard", "question": "Circle with center O. Tangent PQ touches at Q. Chord QA parallel to tangent at P. If ∠PQA = 30°, find ∠QPA. 🔍", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "75°"}, "correct_answer": "C"},
{"id": 56, "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": 57, "difficulty": "hard", "question": "Quadrilateral ABCD circumscribes circle. If AB = 8 cm, BC = 7 cm, CD = 6 cm, find AD. 🔄", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "9 cm"}, "correct_answer": "C"},
{"id": 58, "difficulty": "hard", "question": "Two tangents TP & TQ from T. If OP = 6 cm, PT = 8 cm (O center), find area of triangle OPT. 🔺", "options": {"A": "12 cm²", "B": "18 cm²", "C": "24 cm²", "D": "30 cm²"}, "correct_answer": "C"},
{"id": 59, "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": 60, "difficulty": "hard", "question": "From point P, tangents PA & PB. Line through center O parallel to PA meets PB at Q. If PA = 12 cm, radius = 5 cm, find PQ. 📏", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"}
]
}