The circumference of a circle is calculated by multiplying the diameter (or twice the radius) by π, hence 2πr is the correct formula. This formula is fundamental in circle geometry.

A radius is specifically the segment that connects the center of the circle to any point on its circumference. This is a key geometric definition that underpins many circle properties.

A chord is defined as a line segment whose endpoints lie on the circle. Understanding chords is important for analyzing many circle-related properties.

The radius is the fixed distance from the center to any point on the circle, making it a constant property of the circle. This consistency is a fundamental aspect of circle geometry.

The area of a circle is calculated using the formula πr², where r is the radius. This formula reflects the relationship between the circle's radius and its enclosed area.

In circle geometry, the measure of a central angle directly equals the measure of its intercepted arc. This clear relationship is a basic yet essential concept.

The area of a circle is proportional to the square of its radius. Doubling the radius results in the area being multiplied by 4, hence it quadruples.

An inscribed angle is always half the measure of its intercepted arc. Therefore, an intercepted arc of 100 degrees corresponds to an inscribed angle of 50 degrees.

A tangent is defined as a line that meets the circle at a single point. This point of tangency is the only location where the tangent touches the circle, distinguishing it from other lines.

When the central angle is measured in radians, the arc length is found by multiplying the radius by the angle (rθ). This formula directly relates the two quantities.

A core property of circles is that congruent chords are always the same distance from the center. This reflects the inherent symmetry of the circle.

Since the sector's area is 25% of the circle, its central angle must be 25% of 360 degrees. That calculation yields 90 degrees.

The standard equation of a circle is given by (x - h)² + (y - k)² = r². This form clearly identifies the center (h, k) and radius r of the circle.

A diameter is the longest possible chord in a circle because it passes through the center, making it twice the length of the radius. This distinct property differentiates the diameter from other chords.

According to the inscribed angle theorem, an inscribed angle measures exactly half of its intercepted arc. This is a fundamental rule in circle theorems.

The Power of a Point Theorem describes the relationship between the segments formed when two secants intersect outside a circle. It is a powerful tool for solving problems involving lengths in circle geometry.

Drawing a perpendicular from the center to the chord creates a right triangle with half of the chord as one leg and the radius as the hypotenuse. Using the Pythagorean theorem, the distance is found to be 6 units.

The Alternate Segment Theorem states that the angle between a tangent and a chord is equal to the angle in the alternate segment of the circle. This theorem links tangent properties with inscribed angles.

First, convert 120° to radians (120° equals 2π/3 radians). Then, using the formula for arc length (rθ), the arc length is 9 × (2π/3) = 6π units.

The given equation is in standard form, (x - h)² + (y - k)² = r². Comparing, we see that h = 3, k = -4, and r² = 49, so the radius is √49 = 7. Thus, the center is (3, -4) and the radius is 7.