geometry: chapter 9 Flashcards
circle
a set of points in a plane equidistant from a given point in the plane
center
the given point
radius
any segment that joins the center to a point of the circle
chord
segment whose endpoints lie in the circle
secant
a line that contains a chord
diameter
chord that contains the center of the circle
tangent
line in the plane of a circle that intersects the circle at exactly 1 point
point of tangency
point that the tangent intersects the circle
sphere
set of all points in space at a given distance from a given point
congruent circles
have congruent radii
concentric circles
circles that line in same plane and have the same center
concentric spheres
spheres that have same center
inscribed/ circumscribed polygon
vertices of the polygon lie on the circle
Theorem 9-1
If a line is tangent to a circle then the line is perpendicular to the radius drawn to the point of tangency.
Corollary to Theorem 9-1
Tangents to a circle from a point are congruent
Theorem 9-2
If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle.
common tangent
a line that is tangent to each of two coplanar circles
common internal tangent
intersects the segment joining the centers of the circles
common external tangent
does not intersect the segment joining the centers
tangent circles
coplanar circles that are tangent to the same line at the same point
central angle
angle with its vertex at the center of the circle
minor arc
part of circle that is smaller than half
major arc
part of circle that is greater than half
semicircle
arc with the endpoints at the diameter
postulate 16: Arc Addiction Postulate
the measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs
adjacent arcs
arcs that have exactly one common endpoint
congruent arcs
arcs in the same circle, or congruent circles that have equal measure
theorem 9-3
in the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.
Theorem 9-4
In the same circle or in congruent circles:
1) congruent arcs have congruent chords
2) congruent chords have congruent arcs
Theorem 9-5
A diameter that is perpendicular to a chord bisects the chord and its arc
Theorem 9-6
In the same circle or in congruent circles
1) chords equally distant from the center are congruent
2) congruent chords are equally distant from the center
inscribed angles
an angle whose vertex is on a circle and whose sides contain chords of the circle
Theorem 9-7
The measure of an inscribed angle is equal to half the measure of its intercepted arc
Corollary 1 of theorem 9-7
if two inscribed angles intercept the same arc, then the angles are congruent
corollary 2 of theorem 9-7
an angle inscribed in a semicircle is a right angle
corollary 3 of theorem 9-7
if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary
Theorem 9-8
the measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc
Theorem 9-9
The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs
theorem 9-10
The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.
theorem 9-11
when two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord
Theorem 9-12
When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segments and its external segment.
Theorem 9-13
when a secant segment and a tangent are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment
