Geometry

The sides 9, 12, 15 form a right-angled triangle because 9^2 + 12^2 = 81 + 144 = 225 = 15^2. For a right triangle inscribed in a circle, the hypotenuse is the diameter. The radius is 15 / 2 = 7.5 cm.

The radius r of an inscribed circle in a regular polygon with n sides and side length s is r = (s/2) * cot(180/n). Here n=15, s=1. r = 0.5 * cot(12). The diameter is 2r = cot(12).

In a right triangle with hypotenuse c and altitude h to the hypotenuse, the area is 1/2 * c * h. Also, area is 1/2 * x * y. The hypotenuse c is the diameter of the circumcircle, so c = 2 * 10 = 20. Area = 1/2 * 20 * 8 = 80. Thus, 1/2 * xy = 80, so xy = 160.

In a right-angled triangle, the altitude h to the hypotenuse divides it into segments p and q. The geometric mean theorem states h^2 = p * q. Here, h^2 = 9 * 16 = 144, so h = 12.

In a unit circle, a chord subtending angle 2*theta at the center forms an isosceles triangle with two radii of length 1. Dropping a perpendicular from the center to the chord bisects the angle into theta and the chord into two segments of length sin(theta). Thus, the total length is 2 * sin(theta).

Statement 1 is a standard geometry theorem: the locus of points equidistant from two lines is the angle bisector. Statement 2 is also a standard theorem: a line intersecting concentric circles creates chords of equal length between the outer and inner circles, so AC = BD.

The perpendicular from the center to a chord bisects the chord. This forms a right triangle with the radius (hypotenuse = 25) and half the chord (base = 24). Height^2 = 25^2 - 24^2 = 625 - 576 = 49. Height = 7.

By the Angle Bisector Theorem, the ratio of the sides AB/AC equals the ratio of the segments BD/CD. 16/12 = 4/CD. 4/3 = 4/CD, so CD = 3 cm.

By the Angle Bisector Theorem, AB/AC = BD/DC. Substituting the values, 12/AC = 10/5, which gives AC = 6 cm. The perimeter is AB + AC + (BD + DC) = 12 + 6 + 15 = 33 cm.

The sides are 4, 7.5, and 8.5. Since 4^2 + 7.5^2 = 16 + 56.25 = 72.25 = 8.5^2, it is a right-angled triangle. The inradius r of a right triangle is (a+b-c)/2 = (4 + 7.5 - 8.5)/2 = 3/2 = 1.5 cm.

For similar triangles, the ratio of areas is the square of the ratio of corresponding sides. Area1/Area2 = (BC/EF)^2. 64/121 = (BC/15.4)^2. Taking square roots, 8/11 = BC/15.4. BC = (8 * 15.4) / 11 = 8 * 1.4 = 11.2 cm.

An angle of 180 degrees at the center means the chord is a diameter. The length of the diameter is 36 cm, so the radius is 36 / 2 = 18 cm.

Length of direct common tangent = sqrt(d^2 - (R - r)^2), where d = 25, R = 12, r = 5. Length = sqrt(25^2 - (12 - 5)^2) = sqrt(625 - 49) = sqrt(576) = 24 cm.

The angle subtended by a chord at the center is twice the angle subtended by it on the major arc. 2 * 55 = 110 degrees.