Geometry

The centroid G divides the line segment joining the orthocentre H and circumcentre O in the ratio 2:1. Let O = (x, y). Then (2x + 1*(-5))/3 = 5 and (2y + 1*7)/3 = 5. Solving gives x = 10, y = 4.

In triangle OAB, OA = OB = radius = 6 cm. Thus, it is an isosceles triangle. The base angles OAB and OBA are equal. The sum of angles is 180. Angle OAB + Angle OBA + Angle AOB = 180. 2 * Angle ABO + 114 = 180. 2 * Angle ABO = 66. Angle ABO = 33 degrees.

Arc length = (theta / 360) * 2 * pi * r = (30 / 360) * 2 * 3.14 * 6 = (1 / 12) * 12 * 3.14 = 3.14 cm.

For similar triangles, the ratio of perimeters equals the ratio of corresponding sides. 75 / 50 = x / 20. Solving for x: x = (75 * 20) / 50 = 30.

For a right triangle with hypotenuse h, the area is (1/2) * a * b. Since a^2 + b^2 = h^2, the area is maximized when a = b = h / sqrt(2). Max area = (1/2) * (h/sqrt(2))^2 = h^2 / 4 = 100 / 4 = 25.

Arc length L = 33, radius r = 14. L = r * theta (in radians). 33 = 14 * theta. theta = 33/14 radians. Convert to degrees: (33/14) * (180/pi) = (33/14) * (180 / (22/7)) = (33/14) * (180 * 7 / 22) = (3/2) * 90 = 135 degrees.

Let r be the radius. Distance from center to PQ = sqrt(r^2 - 4^2). Distance to RS = sqrt(r^2 - 8^2). Difference = sqrt(r^2 - 16) - sqrt(r^2 - 64) = 4. Let x = sqrt(r^2 - 64). Then sqrt(x^2 + 48) - x = 4. sqrt(x^2 + 48) = x + 4. x^2 + 48 = x^2 + 8x + 16. 8x = 32. x = 4. r^2 = 4^2 + 64 = 80. r = sqrt(80) = 4*sqrt(5).

A perpendicular from the center to a chord bisects the chord. Using the Pythagorean theorem on the triangle formed by the radius (r), the distance from the center (3 cm), and half the chord (4 cm), we get r^2 = 3^2 + 4^2 = 25, so r = 5 cm. The diameter is 2 * r = 10 cm.

In a right triangle, the radius of the circumcircle is half the hypotenuse. So, hypotenuse = 2 * 3 = 6 cm. Area = (1/2) * base * height = (1/2) * hypotenuse * altitude_to_hypotenuse = (1/2) * 6 * 2 = 6.

Centroid G = ((x1+x2+x3)/3, (y1+y2+y3)/3) = ((4+3+2)/3, (-3-2+8)/3) = (9/3, 3/3) = (3, 1). The distance of (3, 1) from the y-axis is the absolute value of the x-coordinate, which is 3.

The centroid G divides the line segment joining the orthocentre H and circumcentre O in the ratio 2:1. Let H = (x, y). G = (2O + H) / 3. (-2, 3) = (2(4, 1) + (x, y)) / 3. -6 = 8 + x => x = -14. 9 = 2 + y => y = 7. H = (-14, 7).

In a right triangle formed by the radius, tangent, and distance to center, r^2 + 24^2 = 26^2. r^2 = 676 - 576 = 100, so r = 10.

In a circle, the line from the center to the external point bisects the angle between the tangents. Thus, angle APO = 80/2 = 40 degrees. In triangle OAP, angle OAP = 90 degrees (tangent is perpendicular to radius). Therefore, angle AOP = 180 - 90 - 40 = 50 degrees.

In an equilateral triangle inscribed in a circle, the product of the segments of a chord (or secant) is constant. AX is the altitude (5 * sqrt(3) / 2) and XY is the remaining part of the diameter. Using properties of chords, AX * AY = AB * AC = 5 * 5 = 25.