Chapter 10 Test Flashcards
Area formula
A=pi*r^2
Cirumference
C=2pir=pi*diameter
Arc
A piece of the circle; it is made up of two points on a circle and all the points of the circle needed to connect those two points by a single path
Arc measure
The number of degrees it occupies in the circle. Since circles are 360, an arc is at most 360
Central Angle
An angle formed by two radii. It is equal to the measure of the arc it creates
Arc length
The fraction of the circle’s circumference occupied by the arc; expressed in linear units such as feet, centimeters, or inches
Circle
The set of all points equidistant from one central point, called the center
Sector
The region of the circle created by a central angle.
Combined ratio of the sector and central angle
area of sector measure of central angle
_____________ = ________________________
area of circle 360*
Arc length formula
Arc length= measure of the central angle
__________________________ C
360*
Sector area formula
Area of sector= measure of central angle
______________________ (Area of circle)
360*
OR
Area of sector= arc length
______________ (Area of circle)
Circumference
Congruent circles
If two circles are congruent, then they have the same radii
Concentric circles
two or more coplanar circles with the same center
Exterior vs. interior points
• If the distance to a point is less than the radius, then the point is on the interior of the circle
• If the distance to a point is greater than the radius, then the point is on the exterior of the circle
• If the distance to a point is equal to the radius, then the point is on the circle
Distance from a chord to the center
The distance from the center of the chord is the measure of the perpendicular segment from the center to the chord.
Theorem #1
If a radius is perpendicular to a chord, then it bisects the chord
Theorem #2
If a radius bisects a chord that is not a diameter, then it is perpendicular to that chord
(converse of theorem #1)
Theorem #3
The perpendicular bisector of a chord passes through the center of a circle
Equidistance theorem #1
Points on the perpendicular bisector of a segment are equidistant from the endpoints of that segment.
Equidistance theorem #2
If two points are equidistant from the endpoints of a segment, then they are on the perpendicular bisector of that segment.
Theorem #4
If two chords of a circle are equidistant from the center, then they are congruent
Theorem #5
If two chords of a circle are congruent, then they are equidistant from the center of the circle
Minor Arcs
Less than 180*, named with two letters
Major arcs
Greater than 180*, named with 3 letters
Semicircles
Equal to 180*, named with 3 letters
Congruent arcs
Arcs that have the same measure AND are parts of the same circle or congruent circles
Super theorem
In the same or congruent circles…
congruent chordscongruent arc lengthscongruent central angles
Tangent
A tangent line (or segment) intersects a circle at exactly one point. This point is called the point of tangency.
Secant
A secant line (or segment) intersects a circle at exactly two points. The interior part of a secant is a chord.
Postulates
A tangent line is perpendicular to the radius drawn to the point of tangency.
If a line it perpendicular to a radius at its point of contact with the circle, then it is tangent to the circle.
Two-tangent theorem
If two tangent segments are drawn to a circle from an external point, then those segments are congruent
Tangent circles
Circles that intersect each other at exactly one point. There’s external and internal (one inside of the other).
Common tangent
A line that intersects two circles at exactly one point each
Steps for external tangent problems
Step 1: Connect the centers
Step 2: Draw the radii to the points of tangency
Step 3: From the center of the smaller circle, draw a line parallel to the common tangent. This creates a rectangle.
Step 4: Use P.T. and properties of rectangles
Distance between two circles
The distance between two circles lies along the line connecting their centers
Steps for internally tangent problems
Draw them
The vertex of an angle can be in one of FOUR places:
- The center of the circle
• On the circle
• Inside the circle but not at the center - Outside the circle
Inscribed angles
Equal to half the measure of its intercepted arc
Tangent-chord angles
Formed by a tangent and a chord, also equal to the measure of half the intercepted arc
Chord-chord angle
Formed by two chords that intersect inside (but not at the center of) a circle; equal to half the sum of the arcs intercepted by the chord-chord angle and its vertical angle
1/2mA+1/2mB=beta
secant-secant, tangent-tangent, and secant-tangent angles
VERTICES OUTSIDE THE CIRCLE
Equal to half the difference of the intercepted arcs: θ = (1/2)(b - a) BASICALLY θ = (1/2)(bigger arc - smaller arc)
Theorem 1
If two inscribed or tangentchord angles intercept the same or congruent arc(s), then they are congruent
Theorem 2
An angle inscribed in a semicircle is right
Theorem 3
The sum of the measures of a tangent-tangent angle and it’s minor arc is 180*
Inscribed Polygon
A polygon is inscribed in a circle if all of it’s vertices lie on the circle
Circumcenter
The center of a circle circumscribed about a polygon
Circumscribed Polygon
A polygon is circumscribed about a circle if each of its sides is tangent to the circle
Incenter
The center of a circle inscribed in a polygon
Apothem
The segment which joins the center of a regular polygon to the midpoint of one of the sides. If the polygon circumscribes the circle, the apothem is a radius of the circle
Theorem 3
If a quadrilateral is inscribed in a circle, its opposite angles are supplementary
Theorem 4
If a parallelogram is inscribed in a circle, it must be a rectangle
Chord-chord power theorem
If two chords intersect inside a circle, then the product of the measure of the segments of one chord is equal to the product of the measures of the other chord
Tangent-secant theorem
If a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the measure of the tangent segment is equal to the product of the measures of the entire secant segment and its external part
t^2=sxe
Secant-secant power theorem
If two secant segments are drawn from an external point to a circle, then the product of the measures of one secant segment and its external point is equal to the product of the other secant segment and its external part
