Geometry

The distance of a chord from the center is inversely proportional to its length. The longest chord is the diameter (7 cm). The chord closest to the center is the one with the greatest length. GH = 6 cm is the longest chord listed.

By the power of a point theorem, for chords intersecting at P outside the circle, PM * PN = PR * PS. Given PN=3, MN=5, so PM=8. PR=2. Thus 8 * 3 = 2 * PS, so PS = 12. Since PS = PR + RS, 12 = 2 + RS, so RS = 10.

A chord is a line segment whose endpoints both lie on the circle.

A chord is a line segment whose endpoints lie on the circle. Since there are infinite points on a circle, there are an infinite number of chords that can be drawn.

Radius = 10 cm. Chord length = 16 cm. A perpendicular from the center bisects the chord into two 8 cm segments. Using the Pythagorean theorem, distance^2 + 8^2 = 10^2. distance^2 = 100 - 64 = 36. distance = 6 cm.

A chord divides a circle into two segments, which are bounded by two arcs: the minor arc and the major arc.

For a chord of length L subtending angle theta at center, L = 2*r*sin(theta/2). sqrt(2) = 2*r*sin(pi/4) = 2*r*(1/sqrt(2)) = r*sqrt(2). So r = 1.

A perpendicular from the center to a chord bisects the chord. This creates a right triangle with legs of 6 cm (half the chord) and 3 cm (distance from center). The radius is the hypotenuse: sqrt(6^2 + 3^2) = sqrt(36 + 9) = sqrt(45) = 3*sqrt(5).

The distance from the center to a chord of length 18 cm is found using the Pythagorean theorem: sqrt(15^2 - 9^2) = sqrt(225 - 81) = sqrt(144) = 12 cm. Since the chords are on the same side, the distance between them is 12 - 0 = 12 cm, but if they are on opposite sides, it is 12 + 12 = 24 cm. Given the options, 24 cm is the intended answer.

The chord PQ connects (3, -4) and (4, 3). Any point (alpha, beta) on the chord must have its coordinates bounded by the coordinates of the endpoints P and Q. Thus, alpha must be between 3 and 4, and beta must be between -4 and 3.

The tangent to a circle is perpendicular to the radius at the point of contact. This forms a right-angled triangle with the radius, tangent, and distance from the center as the hypotenuse. Using Pythagoras theorem, r^2 + 24^2 = 25^2, so r^2 = 625 - 576 = 49, meaning r = 7 cm.

Since the distance from the center of the circle to point M is 7 cm, which is greater than the circle's radius of 4 cm, point M lies outside the circle. From any point outside a circle, exactly two tangent segments can be drawn to the circle.

The distance from the center to point P is 10 cm, which is greater than the radius of the circle (8 cm). Therefore, point P lies outside the circle. From any point outside a circle, exactly two tangents can be drawn to the circle.