Geometry

Angles subtended by the same arc are equal. Angle CBD = angle CAD = 60 degrees. In triangle BCD, the sum of angles is 180 degrees. Angle BCD = 180 - 60 - 25 = 95 degrees.

Perimeter = 12, so side = 4. In an equilateral triangle, the median is also the altitude. Altitude = (sqrt(3)/2) * side = (sqrt(3)/2) * 4 = 2 * sqrt(3).

The triangle with maximum area inscribed in a semicircle has its base as the diameter (20 cm) and height as the radius (10 cm). Area = 1/2 * base * height = 1/2 * 20 * 10 = 100.

Using Pythagorean theorem, height = sqrt(10^2 - 6^2) = sqrt(64) = 8 cm. Area = 1/2 * base * height = 1/2 * 6 * 8 = 24 cm^2.

For chords intersecting inside a circle, the product of the segments of one chord equals the product of the segments of the other. AE * EB = CE * ED. 5.6 * 10 = 8 * ED. 56 = 8 * ED, so ED = 7.

OA and OB are radii, and angle AOB is 90 degrees, so triangle AOB is a right isosceles triangle. Since AB = 8, its area is half the product of the perpendicular radii, which equals 16 square centimetres.

A chord of length L subtending angle theta at the center has L = 2r*sin(theta/2). Here L = 2*sqrt(2) and theta = 90 degrees. 2*sqrt(2) = 2r*sin(45) = 2r*(1/sqrt(2)) = r*sqrt(2). Thus r = 2.

In a right triangle inscribed in a circle with the hypotenuse as the diameter, the angle opposite the diameter is 90 degrees. Given angle ABC = 30 degrees, the triangle is a 30-60-90 triangle. AC is opposite the 30-degree angle. AC = Diameter * sin(30) = Diameter * 0.5. 5 = Diameter * 0.5, so Diameter = 10.

In a circle, a chord of length L subtends angle theta at the center. L = 2R sin(theta/2). 2*sqrt(3) = 2R sin(60). sqrt(3) = R * (sqrt(3)/2), so R = 2. Area = pi * R^2 = pi * 2^2 = 4pi.