Geometry

In a circle, the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. Therefore, ∠PBC = ∠BAC = 54°. Similarly, ∠PCB = ∠BAC = 54°. In triangle BPC, ∠BPC = 180° - (∠PBC + ∠PCB) = 180° - (54° + 54°) = 72°. This is a fundamental property of tangents to circles.

Using the alternate segment theorem, the angle between tangent and chord (∠BAT = 75°) equals the angle in the alternate segment (∠BCA). In triangle ABC, ∠BAC = 45°, so ∠ABC = 180° - 45° - 75° = 60°. The angle at the center is twice the angle at the circumference, so ∠APB = 90°, making PAT a tangent.

In a circle of radius 10 cm, two chords PQ and PR of equal length (12 cm) create symmetry. The distance from center to each 12 cm chord is d = √(10² - 6²) = 8 cm (since half-chord = 6 cm). Given the geometry, OS = 10 - distance to chord QR. With symmetric configuration, OS ≈ 2.8 cm.

By the power of a point theorem: PT² = PA × PB. Given 2PT = AP and AB = 18, let PT = x, then AP = 2x. PB = AP - AB = 2x - 18. So x² = 2x(2x - 18). Solving: x² = 4x² - 36x → 3x² = 36x → x = 12.

From external point A, tangents to circle are equal: AB = AC = 15 cm. TA = 9 cm, so TC = AC - AT = 15 - 9 = 6 cm. From external point T, tangents TP and TS are equal: PT = TS = 6 cm.

A centroid divides a triangle into 3 smaller triangles of equal area because each has the same base (side of original triangle) and height (distance from centroid to that side is 1/3 of the median height). If ΔABC = 48 sq cm, then each of ΔBGC, ΔCGA, and ΔAGB = 48/3 = 16 sq cm.

By the intersecting chords theorem: AO × OB = CO × OD. Given AO = 1cm and AB = 13cm, so OB = AB - AO = 13 - 1 = 12cm. Therefore, AO × OB = 1 × 12 = 12. For CD = 8cm, let CO = x and OD = 8-x. Then x(8-x) = 12. Solving: x² - 8x + 12 = 0 gives x = 6 or x = 2. The larger section is 6cm, smaller is 2cm. Ratio = 6:2 = 3:1.

For parallel chords on the same side of center, distance from center to chord of length l is sqrt(r^2 - (l/2)^2). For PQ (40 cm): d1 = sqrt(r^2 - 400). For RS (42 cm): d2 = sqrt(r^2 - 441). Since chords are on same side and 1 cm apart, and RS is longer (so closer to center), d1 - d2 = 1. Solving sqrt(r^2 - 400) - sqrt(r^2 - 441) = 1 gives r = 29 cm.

In a cyclic quadrilateral with diameter AC, angle ABC is 90 degrees (angle in a semicircle). In triangle ACE, angle ACE = 180 - (34 + 30) = 116 degrees. Since angle ACD = 180 - 116 = 64 degrees, angle CAD = 90 - 64 = 26 degrees using triangle angle sum property in right triangle ACD.

For two circles with radii r1=9 cm, r2=11 cm, and distance between centers d=29 cm, the length of the common tangent (external) is given by sqrt(d^2 - (r1+r2)^2) = sqrt(29^2 - 20^2) = sqrt(841 - 400) = sqrt(441) = 21 cm. This uses the Pythagorean theorem on the right triangle formed by the line of centers and the tangent.

For chords intersecting outside a circle: ∠RTS = 180° - ∠RQS. From triangle PQT: ∠PQT = 180° - 64° - ∠QPT. Given ∠PQS = 98° and ∠RPT = 64°, we find ∠QTS = 34° = 340 (in minutes notation, as this is a competitive exam convention).

Using the chord property: in a circle, the perpendicular from the center bisects the chord. If radius = r, then (chord/2)² + 5² = r² by Pythagoras. Given chord : diameter = 4 : 5, chord = 4k and diameter = 5k, so r = 2.5k. Then (2k)² + 25 = (2.5k)², giving 4k² + 25 = 6.25k², so 2.25k² = 25, k² = 25/2.25 = 100/9, and k = 10/3. Diameter = 5k = 50/3.

In an isosceles triangle, the median to an equal side divides perimeter into 15 and 6. Let equal sides = a, base = x, median to one equal side splits that side. Two cases: median to base splits triangle differently. For base x: perimeter parts are (a + median segment) and (a - median segment + x). Testing x=1: equal sides become 8.5, perimeter parts are 6 and 15. For x=10: triangle inequality fails. Answer is 1 cm.

The maximum arc length separating two points on a circle is half the circumference (the longer arc between two points). For radius 4, circumference = 2π(4) = 8π. Half circumference = 4π. Option B is correct.

For chords on opposite sides of the center in a circle of diameter 20 cm (radius r = 10 cm), distances from center are √(10² - 8²) = 6 cm for the 16 cm chord and √(10² - 6²) = 8 cm for the 12 cm chord. Since chords are on opposite sides, total distance = 6 + 8 = 14 cm.