Parts of a circle:
- Radius: The distance from the center of the circle to its outer rim.
- Chord: A line segment whose endpoints are on a circle.
- Diameter: A chord that passes through the center of the circle.
- Secant: Intersects the circle in 2 places
- Tangent: intersects with the circle in one place, forms a right angle when connected to a radius or diameter.
Central angles
Inscribed angles
Inscribed Quadrilaterals
Arcs formed by parallel chords
When a circle has two chords that are parallel, the arcs between the two chords are always congruent. With a transversal drawn in, the parallel lines make two inscribed angles, which are alternate interior to each other and congruent inscribed angles intercept congruent arcs.
Angles formed by intersecting chords
Segments formed by intersecting chords
Secants- Intersects with a circle in two places
Tangent- touches a circle in exactly one place. When connected to a radius it forms a right angle.
Tangent- touches a circle in exactly one place. When connected to a radius it forms a right angle.
Angles formed by Secants and tangents
Just like inscribed angles are half the measure of their arcs
Angles formed by secants and tangents are half of the difference between the measure of their arcs.
Example (125 -27)/2 = x (Major arc - Minor arc)/2 = angle measure
x = 49 degrees
Angles formed by secants and tangents are half of the difference between the measure of their arcs.
Example (125 -27)/2 = x (Major arc - Minor arc)/2 = angle measure
x = 49 degrees
Solving for missing segments in secants and tangents
When two secants meet outside a circle. Missing lengths can be found using:
Whole secant x outside part = Whole secant x outside part Below 6 x 4 = 8 x 3 |
When a secant meets a tangent, the same rule applies.
However, the whole tangent is also the outside part So: Whole x outside = tangent squared Below: 16 x 7 = x(squared) 112 = x squared √112 = x or 10.58 |
Equation of a circle
A circle is a round figure where all points are equal distance from it's center.
Because of this a circle with it's center at the origin has an equation of X^2 + Y^2 = radius^2
Because of this a circle with it's center at the origin has an equation of X^2 + Y^2 = radius^2
Equation of a circle - not centered at the origin
A circle that is not centered as the origin has a slight difference in it's equation.
Example:
This circle has it's center at ( 3 , 1 ) and has a radius of 2.
To simplify how the equation changes we need to think:
"How can I get this circle back to the origin?"
To do this we would need to move it left 3 (-3) and down 1 (-1)
(x -3)^2 + (y - 1)^2 = 4
Remember: Swap the signs to get the location of the center.
Watch the following video for a walk through.
Example:
This circle has it's center at ( 3 , 1 ) and has a radius of 2.
To simplify how the equation changes we need to think:
"How can I get this circle back to the origin?"
To do this we would need to move it left 3 (-3) and down 1 (-1)
(x -3)^2 + (y - 1)^2 = 4
Remember: Swap the signs to get the location of the center.
Watch the following video for a walk through.