Main content

### Math

## Common Core Math

# High School: Geometry: Circles

Prove that all circles are similar.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

- Challenge problems: circumscribing shapes
- Challenge problems: Inscribed angles
- Challenge problems: Inscribed shapes
- Challenge problems: radius & tangent
- Circles glossary
- Determining tangent lines: angles
- Determining tangent lines: lengths
- Inscribed angle theorem proof
- Inscribed angle theorem proof
- Inscribed angles
- Inscribed angles
- Inscribed shapes
- Inscribed shapes: angle subtended by diameter
- Inscribed shapes: find diameter
- Inscribed shapes: find inscribed angle
- Proof: perpendicular radius bisects chord
- Proof: radius is perpendicular to a chord it bisects
- Proof: Radius is perpendicular to tangent line
- Proof: Right triangles inscribed in circles
- Proof: Segments tangent to circle from outside point are congruent
- Tangents of circles problem (example 1)
- Tangents of circles problem (example 2)
- Tangents of circles problem (example 3)
- Tangents of circles problems

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

- Geometric constructions: circle-inscribed equilateral triangle
- Geometric constructions: circle-inscribed regular hexagon
- Geometric constructions: circle-inscribed square
- Geometric constructions: triangle-circumscribing circle
- Geometric constructions: triangle-inscribing circle
- Inscribed quadrilaterals proof
- Solving inscribed quadrilaterals

Construct a tangent line from a point outside a given circle to the circle.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

- Arc length as fraction of circumference
- Arc length from subtended angle
- Arc length from subtended angle: radians
- Arc measure
- Arc measure with equations
- Arcs, ratios, and radians
- Area of a sector
- Challenge problems: Arc length (radians) 1
- Challenge problems: Arc length (radians) 2
- Challenge problems: Arc length 1
- Challenge problems: Arc length 2
- Degrees to radians
- Deriving the area of a sector
- Finding arc measures
- Finding arc measures with equations
- Intro to arc measure
- Intro to radians
- Radians & arc length
- Radians & degrees
- Radians & degrees
- Radians as ratio of arc length to radius
- Radians to degrees
- Subtended angle from arc length