Lesson 3 • 50 min read
Circles
In This Lesson
1Circle Equation
Standard Form
(x - h)² + (y - k)² = r²
(h, k)
Center
r
Radius
r²
Radius squared
General Form
x² + y² + Dx + Ey + F = 0
To convert to standard form:
- Group x and y terms
- Complete the square for both
- Write in standard form
Examples
Circle with center (3, -2), radius 5:
(x - 3)² + (y + 2)² = 25
Circle at origin, radius 4:
x² + y² = 16
2Central Angles
Definition
A central angle is an angle whose vertex is at the center of the circle.
Central Angle = Intercepted Arc
The measure of a central angle equals the measure of its intercepted arc.
Minor Arc
Arc less than 180°. Named with two letters (e.g., AB).
Major Arc
Arc greater than 180°. Named with three letters (e.g., ACB).
Example:
If central angle AOB = 60°
Then arc AB = 60°
And major arc ACB = 360° - 60° = 300°
3Inscribed Angles
Definition
An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords.
Inscribed Angle = ½ × Intercepted Arc
The measure of an inscribed angle is half the measure of its intercepted arc.
Important Theorems
- •Inscribed in Semicircle: An angle inscribed in a semicircle is always 90°
- •Same Arc: Inscribed angles that intercept the same arc are equal
- •Opposite Angles: In an inscribed quadrilateral, opposite angles are supplementary (add to 180°)
Example:
If arc PQ = 80°
Inscribed angle PRQ = 80° ÷ 2 = 40°
Central angle POQ = 80° (equals the arc)
4Arc Length & Sector Area
Arc Length
L = (θ/360°) × 2πr
L = arc length
θ = central angle (in degrees)
r = radius
Sector Area
A = (θ/360°) × πr²
A = sector area
θ = central angle (in degrees)
r = radius
Using Radians
Arc Length
L = rθ
Sector Area
A = ½r²θ
Example:
Circle with radius 10 cm, central angle 72°
Arc length = (72/360) × 2π(10) = 4π ≈ 12.57 cm
Sector area = (72/360) × π(10)² = 20π ≈ 62.83 cm²
5Tangent & Secant Lines
Tangent Line
A line that touches the circle at exactly one point (point of tangency).
Tangent ⊥ Radius at point of tangency
Secant Line
A line that intersects the circle at exactly two points.
A chord is part of a secant
Important Theorems
Two Tangents from External Point
The two tangent segments are equal in length.
Tangent-Secant Angle
Angle = ½ × (difference of intercepted arcs)
Secant-Secant Angle
Angle = ½ × (difference of intercepted arcs)
Power of a Point Theorems
| Configuration | Formula |
|---|---|
| Two Chords | a × b = c × d |
| Two Secants | (whole₁)(external₁) = (whole₂)(external₂) |
| Tangent-Secant | tangent² = (whole)(external) |