Math Ch. 10 Theorems and Vocab
If two inscribed or tangent - chord angles intercept the same arc, then they are congruent.
Theorems?
If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruent.
Theorems?
The sum of the measures of a tangent-tangent angle and its minor intercepted arc is 180 degrees. measure of angle M + measure of arc NP = 180. MNOP is a kite. Angles add to 360. Measure of angle M + measure of angle O = 180.
Theorems?
each side of the polygon is tangent to the circle.
a polygon is circumscribed about a circle if..
each side of the polygon is tangent to the circle
A circle is inscribed in a polygon if
all of the polygon's vertices are on the circle.
A cirlce is circumscribed about a polygon if...
all of its vertices are on the circle
A polygon is inscribed in a cirlce if...
arcs ≅ --> central angles ≅
Describe the relationship in each diagram
arcs ≅ --> chords ≅
Describe the relationship in each diagram
central angles ≅ --> arcs ≅
Describe the relationship in each diagram
central angles ≅ --> chords ≅
Describe the relationship in each diagram
chords ≅ --> arcs ≅
Describe the relationship in each diagram
chords ≅ --> central angles ≅
Describe the relationship in each diagram
If 2 points are equidistant from endpoints of a seg, then perpendicular bisector.
If 2 points are equidistant from endpoints of a seg, ----
If 2 tangent segs are drawn to a circle from the same exterior point, then they are congruent.
If 2 tangent segs are drawn to a circle from the same exterior point....
If a radius bisects a chord, that is not a diameter, then it is perpendiuclar to the chord
If a radius bisects a chord, that is not a diameter,
If a radius of a circle is perpendicular to a chord, then it bisects that chord
If a radius of a circle is perpendicular to a chord, ----
if central angles congruent <=> arcs congruent <=> chords congruent
If same circle or congruent angles..
sometimes
If two arcs have the same measure, then they are (A/S/N) congruent
If two chords of a circle are congruent, then they are equidistant form the center of the circle
If two chords of a circle are congruent....
If two chords of a circle are equidistant from the center, then they are congruent.
If two chords of a circle are equidistant from the center...
A central angle
Is an angle whose vertex is at the center of a circle
90
Measure of AB
120
Measure of AB?
the major arc
Measure of AXB
or circle R is inscribed in the pentagon. Moreover, R is the incenter of LMNOP
Pentagon LMNOP is circumscribed about circle R,
Circle O is circumscribed about the quad. Moreover, O is the circumcenter of ABCD
Quad ABCD is inscribed in circle O,
180
measure of semicircle
center of the polygon. The center is equidistant to the sides of the polygon.
The center of a circle is called the..
circumcenter of the polygon. A circumcenter is equidistant to the vertices of the polyogn.
The center of the circle is called the
The perpendicular bisector of a chord passes through the center of the circle
The perpendicular bisector of a chord .....
(tangent segment)^2 = external segment x secant segment a^2 = c (b+c)
Theorem for Tangent - secant
two tangents segments from the same external point are congruent.
Theorem for tangent - secant
If a parallelogram is inscribed in a circle, it must be a rectangle. reasoning: 1. oppostie angles of a parallelogram are congruent 2. opposite angles of an inscribed quad are supp.
Theorems?
If a quadrilateral is inscribed in a cirlce, its opposite nagles are supplementary. measure of angle BAD = 1/2 of arc BC Measure of angle BCD = 1/2 of arc BAD
Theorems?
If an angle is inscribed in a semicricle, then it is a right angle
Theorems?
- opposite angles of a quadrilateral inscribed in a circle are supplementary
angles and quadrilaterals
- angle inscribed in a semicircle is a right angle
angles and semicircles
- measure of arc is same as measure of central angle
arcs and angles
area sector - area of triangle
area of circle segment
(x / 360) x area
area of sector
x = (arc of measure AB)
central angle
x = 1/2 (arc of mAB + arc of mCD)
chord chord angle
chords equidistant from the center are congruent
chords and center
chord through point of tangency perpendicular to tangent passes through center
chords and tangents
x = arc measure. (x / 360) x circumference
length of arc
arc
consists of two points on a circle and all points on the circle needed to connect the points by a single path.
same measure congruent, or same circle
definition of congruent arcs
central angles ≅ <--> chords ≅ <--> arcs ≅
describe relationship between central angles, chords, and arcs
diameter is longest chord
diameter and chord
A = 𝛑r²
formula of area
C = 𝛑d
formula of circumference
if a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle
if a line is perpendicular to a radius at its outer endpoint....
Perpendicular line segment
line from the center of the circle to a chord
if line perpendicular to radius at outer endpoint => tangent
line perpendicular to radius
x = 1/2 (arc of measure AB)
inscribed angle
semicircle
is an arc whose endpoints are the endpoints of a diameter
Minor arc
is an arc whose points are on or between the sides of a central angle (less than 180)
Major arc
is an arc whose points are on or outside of a central angle (more than 180)
if a line from center is perpendicular to chord <=> it bisects the chord
perpendicular and chords
perpendicular bisector of a chord passes through center of the circle
perpendicular bisector and center
x = 1/2 (arc of mAB - arc of mCD)
secant secant angle
x = 1/2 (mAB - mAC)
secant tangent angle
if tangent => perpendicular to radius
tangent and radius
x = 1/2 (arc of measure AB)
tangent chord
x = 1/2 (arc of mACB - arc of mAB)
tangent tangent angle
tangent-tangent angle and its intercepted arc are supplementary
tangent tangent angles and intercepted arc
Product of segments of one chord = product of segments of other chord ad = bc
theorem for chord-chord
external segment x secant segment = external segment x secant segment b(a+b) = d(c+d)
theorem for secant - secant
measure of angle x. = 1/2 sum intercepted arcs measure of angle x = 1/2 (measure of arc AB + measure of arc CD)
vertex inside circle
measure of angle x = 1/2 measure intercepted arc measure of angle x = 1/2 measure of arc AB
vertex on circle
measure of angle x = 1/2 difference intercepted arcs measure of angle x = 1/2 (measure of arc AB - measure of arc CD)
vertex outside circle
a tangent line is perpendicular to a radius drawn to the point of tangency
when is a tangent line perpendicular to its radius?