TRIGONOMETRY QTR4

• Radius: Distance from the center to any point on the circle.

• Diameter: The length of a straight line passing through the center connecting two points on the circle. Diameter = 2 x Radius.

• Circumference: The distance around a circle.

• Arc: A part of the circle between two points.

• Chord: A line segment whose endpoints lie on the circle.

• Tangent: A line that touches the circle at exactly one point.

• Sector: A portion of the circle defined by two radii and the arc between them.

• Segment: A region of the circle bounded by a chord and the arc connecting its endpoints.

Circumference of a Circle

• Definition: The circumference is the total distance around a circle.

• Measurement: Use a string to measure the circumference.

• Relationship with Diameter: The circumference is proportional to the diameter (C = πd).

• Calculation Example:

  • If radius r = 3 m, then:

    • Diameter d = 2 * r = 6 m

    • C = π * d = 3.14 * 6 = 18.84 m

Area of a Circle

• Formula: A = πr²

• Intuition: A circle can be cut into equal sectors that can form a parallelogram. Height (h) = radius (r) and base (b) = 1/2 circumference (C).

• Derivation: A = bh = (1/2)(2πr)(r) = πr².

Measuring Arcs and Angles

• Minor Arc: The arc corresponding to the central angle that is less than 180 degrees.

• Major Arc: The arc corresponding to the central angle that is more than 180 degrees.

• Arc Length Formula: The ratio of the arc length to circumference is equal to the ratio of the measure of the arc to 360 degrees.

• Central Angle: An angle with its vertex at the center of the circle.

  • Theorem: The central angle is twice the measure of the inscribed angle intercepting the same arc.

Inscribed Angles

• Definition: An inscribed angle has its vertex on the circle and sides as chords of the circle.

• Inscribed Angle Theorem: Measure of an inscribed angle is half the measure of its intercepted arc.

• Special Case: An angle inscribed in a semicircle is a right angle.

• Examples:

• Measure of angles using established theorems related to inscribed angles and quadrilaterals.

Tangents to a Circle

• Definition: A tangent is a line that intersects the circle at exactly one point.

• Common Tangents: Lines tangent to multiple circles simultaneously. Types include:

  • 0, 1, or 2 common tangents based on circle arrangement.

• Theorem: If a line is tangent to a circle, it is perpendicular to the radius at the point of tangency.

• Example Problem:

  • Use the Pythagorean theorem in problems involving tangents to find unknown lengths.

Summary of Theorems

• Congruent Tangent Segments: Tangent segments from the same external point to a circle are congruent.

• Key Example: If two tangent segments are drawn from external point A to points on the circle, they are equal, i.e., AB = AC if both are tangents to the same circle.