Geometry Chapter 6 Flashcards
Circumscribed polygon
A polygon all of whose sides are tangents to a circle.
Principle 13: In the same or congruent circles, chords that are equally distant from the center…
Are congruent
If OE=OF, OE is perpendicular to AB, and OF is perpendicular to CD, then AB=CD
A tangent is perpendicular to…
the radius drawn to the point of contact.
If AB is a tangent to circle O at P, and OP is drawn, then AB is perpendicular to OP.
In the same or congruent circles, congruent inscribed angles have…
congruent intercepted arcs.
If angle 1=angle 2, then arc BC=DE
Major Arc
An arc that is greater than a semicircle.
Angles inscribed in the same or congruent arcs are…
congruent
if angle C and D are inscribed in arc ACB then angle C=angle D
Central Angle
Angle formed by two radii
Minor Arc
An arc that is less than a semicircle
A diameter perpendicular to a chord
bisects the chord and its arcs
If CD is perpendicular to AB, then CD bisects line segments AB, arc AB, and arc ACB
Circles Outside Each Other
Circles O and O’ are entirely outside each other. The common internal tangents, AB and CD meet at P. If the circles are unequal, their common external tangents, EF and GH if extended, meet at P’. The line of centers OO’ passes through P and P’. Also, AB=CD and EF=GH
In the same or congruent circles, inscribed angles having congruent intercepted arcs are…
Congruent
If arc BC=DE, then Angle 1=angle 2
Opposite angles of an inscribed quadrilateral…
are supplementary
If ABCD is an inscribed quadrilateral, A is the supplement of C.
Principle 7: In the same or congruent circles…
Congruent arcs have congruent central angles
If AC=CB then angle 1=angle 2
Inscribed Circle
A circle to which all the sides of a polygon are tangents.
Overlapping Circles
Circles O and O’ overlap. Their common chord is AB. IF the circles are unequal, their equal common external tangents CD and EF meet at P. The line of centers OO’ is the perpendicular bisector of AB and if extended, passes through P
Secant of a circle
A line that intersects the circle at two points.
Inscribed polygon
A polygon all of whose sides are chords of a circle.
Tangents to a circle from an outside point…
are congruent
if AP and AQ are tengent to circle O at P and Q, then AP=AQ
Two circles are equal…
If their radii are equal in length
Circles tangent externally
Circles O and O’ are tangent externally at P. AB is the common internal tangent of both circles. The line of centers OO’ passes through P, is perpendicular to AB, and is equal in length to the sum of the radii, R+r. Also AB bisects each of the common external tangents, CD and EF.
If a chord divides a circle into two parts…
then it is a diameter.
If ACB=ADB then AB is a diameter
an angle inscribed in a semicircle…
is a right angle
since angle C is inscribed in semicircle ACD, angle C=90 degrees.
An angle formed by two secants intersecting outside a circle is measured by
one half the difference of the intercepted arcs.
In the same or congruent circles….congruent central angles have…
congruent arcs.
if angle 1=angle 2 then AC=CB
A line is tangent to a circle if…
It is perpendicular to a radius at its outer end.
If AB is perpendicular to radius OP at P, then AB is tangent to circle O.
The Line of centers of two circles is…
the line joining their centers.
OO’ is the line of centers of circles O and O’.
An angle formed by two intersecting chords is measured by…
one half the sum of the intercepted arcs
An angle formed by a tangent and a secant intersecting outside a circle is measured by…
one half the difference of the intercepted arcs.
Circumscribed Circle
A circle passing through each vertex of a polygon.
Circle o is a circumscribed circle of quadrilateral ABCD
An angle formed by two tangents intersecting outside a circle is measured by…
One half the difference of the intercepted arcs.
Principle 1: A diameter divides a circle…
Into two equal parts.
diameter AB divides circle O into two congruent semicircles.
An inscribed angle is measured by…
one half its intercepted arc.
Principle 9: In the same or congruent circles, congruent arcs have…
Congruent chords
If arcs AB=AC, then line segments AB=AC
A point is outside, on or inside a circle according…
to wheter its distance from the center is greater than, equal to, or smaller than the radius.
F is outside circle O, since FO is greater in length than a radius. E is inside circle O since EO is smaller in length than a radius. A is on circle O since AO is a radius
A line passes through the center of a circle…
if it is perpendicular to a tangent at its point of contact.
If AB is tangent to circle O at P, and CP is perpendicular to AB at P, then CP extended will pass through the center O.
Principle 12: In the same or congruent circles, congruent chords…
are equally distant from the center
if segment AB=CD, if OE is perpendicular to AB, and if OF is perpendicular to CD, then OE=OF
A central angle is measured by…
its intercepted arc.
Another way to state this: A central angle has the same number of degrees as the arc it intercepts.
An angle formed by a tangent and a chord is measured…
by one half its intercepted arc
Parallel lines intercept…
congruent arcs on a circle
If AB is parallel to CD, then arc AC=BD. If tangent FG is parallel to CD, then arc PC=PD
The segment from the center of a circle to an outside point…
bisects the angle between the tangents from the point to the circle.
OA bisects angle PAQ if AP and AQ are tangents to circle O.
Circles Tangent Internally
Circles O and O’ are tangent internally at P. AB is the common external tangent of both circles. The line of centers OO’ if extended passes through P, is perpendicular to AB, and is equal in length to the difference of the radii, R-r.
The length of a tangent from a point to a circle
is the length of the segment from the given point to the point of tangency.
PA is the length of the tangent from P to circle O.
