Geometry

Circumference = 2πr = 88 cm, so r = 14 cm. A chord that subtends 90° at the circumference subtends 180° at the center, making it a diameter. Therefore XY = 2r = 2 × 14 = 28 cm.

For any chord, the angle at the center is twice the angle at the circumference standing on the same arc. Since the angle on the major arc is 55°, the reflex angle at the center = 2 × 55° = 110°.

If an angle subtended by a chord on the circumference is 90°, then that chord is the diameter of the circle (Thales' theorem). The chord PQ is therefore the diameter. Given radius = 15 cm, diameter = 2 × 15 = 30 cm. So length of PQ = 30 cm. Options A, B are incorrect calculations, and option C is just the radius length.

When the sum of radii equals the distance between centers (r1 + r2 = C1C2), the circles touch externally at exactly one point. In this configuration, there are 3 common tangents: 2 direct tangents (one above and one below both circles) and 1 transverse tangent at the point of contact. Option A (1 tangent) would mean complete overlap, while option D (4 tangents) requires the circles to be separate without touching.

By the Angle Bisector Theorem, AD divides BC proportionally to the adjacent sides: BD/DC = AB/AC. Given AB = 8.4 cm, AC = 5.6 cm, and DC = 2.8 cm, we get BD/2.8 = 8.4/5.6 = 3/2. Therefore BD = 2.8 × (3/2) = 4.2 cm. The full length BC = BD + DC = 4.2 + 2.8 = 7 cm.

By the Angle Bisector Theorem, the bisector divides the opposite side proportionally to the adjacent sides: BD/CD = AB/AC. Given AB = 10 cm and AC = 14 cm, we have BD/CD = 10/14 = 5/7. Therefore, BD : CD = 5 : 7. The ratio is directly obtained from the theorem without needing absolute lengths.

In a triangle, the centroid divides the median in a 2:1 ratio, with the longer segment being from the vertex. AP is the distance from vertex A to the centroid P. Since the centroid divides median AD in the ratio AP:PD = 2:1, and AP = 14 cm, then PD = AP/2 = 14/2 = 7 cm.

For two concentric circles with radii R = 20 cm (larger) and r = 16 cm (smaller), a chord of the larger circle tangent to the smaller circle will be perpendicular to the radius at the point of tangency. Using the Pythagorean theorem: half the chord length = √(R² - r²) = √(400 - 256) = √144 = 12 cm. Therefore, full chord length = 2 × 12 = 24 cm (option B).

For a triangle with sides 16 cm, 20 cm, 24 cm and an inscribed circle, the tangents from each vertex to the circle are equal. Let the tangent lengths be x, y, z from vertices A, B, C respectively. Then x+y = 16, y+z = 20, x+z = 24. Solving: Adding all three: 2(x+y+z) = 60, so x+y+z = 30. Therefore x = (x+y+z) - (y+z) = 30 - 20 = 10 cm. AD is the tangent from A, which equals x = 10 cm (option A).

By the tangent-secant power theorem: MT^2 = MA × MB. Given MT = 12, MA = 8, we get 144 = 8 × MB, so MB = 18. Since AB = MB - MA, AB = 18 - 8 = 10cm. Option A (16cm) would be the whole secant MB. Option C (12cm) is the tangent length. Option D (9cm) doesn't match the calculation.

When two circles touch externally and a direct common tangent touches them at X and Y, the angle formed at the point of contact (Z) with the tangent points X and Y is always 90 degrees. This is because the radius to the point of tangency is perpendicular to the tangent. Since both radii ZX and ZY are perpendicular to the tangent XY, angle XZY = 90 degrees. This matches option A.

An angle subtended by a chord at the circumference is 90° only if the chord is the diameter of the circle. This is Thales' theorem: the angle in a semicircle is always 90°. Since chord AB subtends 90° on the circumference, AB must be the diameter. Therefore, the radius = diameter/2 = 16/2 = 8 cm. This matches option A.

A tangent is perpendicular to the radius at the point of contact. Triangle OTP is right-angled at T, so by Pythagoras: PT² = OP² - OT² = 17² - 8² = 289 - 64 = 225, giving PT = 15 cm.

Since triangle ABC is isosceles with AB=AC, the base angles ∠ABC and ∠ACB are equal. Given ∠ABC=55°, we have ∠ACB=55° and ∠BAC=180°-55°-55°=70°. In an isosceles triangle, the median to the base is also the angle bisector of the vertex angle, so AD bisects ∠BAC. Therefore, ∠BAD=70°/2=35°.

Quantity I: In triangle with sides 15, 9, 12, the median to shortest side (9cm) is calculated using Apollonius theorem: m = √(2a² + 2b² - c²)/2 = √(2×225 + 2×144 - 81)/2 = 13.5cm. Quantity II: Area = √(18×3×6×9) = 54cm² using Heron's formula. For equal area rectangle with length 9, width = 6, so perimeter = 2(9+6) = 30cm. Since 13.5 < 30, Quantity II > Quantity I.