Geometry

In a 30-60-90 triangle, the side opposite the 30 degree angle is half the length of the hypotenuse. Since the hypotenuse is 10, the side is 10 / 2 = 5.

The angle between the tangents and the angle subtended by the chord at the center are supplementary. 180 - 72 = 108 degrees.

A circle tangent to the x-axis has a radius equal to the absolute value of the y-coordinate of its center. Center is (-3, -4), so radius = |-4| = 4.

The arc length formula is (theta/360) * 2 * pi * r. With theta=60 and r=21, length = (60/360) * 2 * (22/7) * 21 = (1/6) * 2 * 22 * 3 = 22 cm.

Arc length = (theta / 360) * Circumference. 22 = (18 / 360) * C. 22 = (1 / 20) * C. C = 22 * 20 = 440 cm.

In a right triangle, hypotenuse^2 = base^2 + height^2. 13^2 = base^2 + 12^2. 169 = base^2 + 144. base^2 = 25, so base = 5 cm.

Using the Pythagorean theorem, the hypotenuse is sqrt(12^2 + 3.5^2) = sqrt(144 + 12.25) = sqrt(156.25) = 12.5 inches.

Area of sector = (1/2) * arc_length * radius = 0.5 * 2.2 * 3.5 = 3.85 cm^2.

Let legs be a and b. a + b = 28 and a - b = 4. Adding gives 2a = 32, so a = 16. Then b = 12. Hypotenuse = sqrt(16^2 + 12^2) = sqrt(256 + 144) = sqrt(400) = 20.

Hypotenuse = sqrt(5^2 + 6^2) = sqrt(25 + 36) = sqrt(61). sqrt(61) is approximately 7.81.

In a right triangle, a^2 + b^2 = c^2. Given base=12, hypotenuse=13, height^2 = 13^2 - 12^2 = 169 - 144 = 25. Height = 5. Area = 1/2 * base * height = 1/2 * 12 * 5 = 30.

For tangents PQ and PR from point P to a circle with center O, the radius OQ is perpendicular to tangent PQ. Thus, triangle OQP is a right-angled triangle with hypotenuse OP. By Pythagorean theorem, OQ^2 + PQ^2 = OP^2.

In a right-angled isosceles triangle with hypotenuse h, the legs are equal (say x). By Pythagoras, x^2 + x^2 = h^2, so 2x^2 = 12^2 = 144, meaning x^2 = 72. The area is (1/2) * base * height = (1/2) * x * x = (1/2) * 72 = 36.

In a right triangle with sides 15 and 17, the third side is sqrt(17^2 - 15^2) = sqrt(289 - 225) = sqrt(64) = 8. The radius of the inscribed circle in a right triangle is r = (a + b - c) / 2, where a and b are legs and c is the hypotenuse. r = (8 + 15 - 17) / 2 = 6 / 2 = 3 cm.