Chapter 11 Flashcards
chord
a segment whose endpoints are points on a circle
secant
a line in the plane of a circle that intersects the circle
point of tangency
a point on the line that intersects the circle in exactly one point
Theorems 11.1 and 11.2
“properties of tangents”
If a line is tangent to a circle, then it is perpendicular to the radius drawn at the point of tangency
In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
tangent segment
touches a circle at one of the segment’s endpoints and lies in the line that is tangent to the circle at that point
Theorem 11.3
If two segments from the same point outside a circle are tangent to the circle, then they are congruent.
minor arc
if a measure of an angle is less than 180, then all the points on that angle form a minor arc
arcs are denoted by A⌒B
(an angle that has a vertex at the very center of the circle)
major point
all the other points that do not lie on A⌒B, form a major arc (these require 3 points to denote)
measure of a major arc
the difference of 360` and the measurement of the related minor arc
semicircle
an arc whose central angle measures 180`, named by three points
Postulate 16 “Arc Addition Postulate”
The measure of an arc formed by two adjacent arcs is the sum of the measured of the two arcs.
congruent circles
two circles that have the same radius
congruent arcs
two arcs of the same circle or of congruent circles if they have the same measure.
arc length
a portion of the circumference of a circle.
You can write a proportion to find arc length:
arc length of A⌒B = mA⌒B
———– of 2(3.14)(radius)
360`
Arc length
In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360`
Theorem 11.4 “diameter perpendicular to a chord”
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc
Theorem 11.5 “chord perpendicular to another chord”
If one chord is perpendicular bisector of another chord, then the first chord is a diameter.
Theorem 11.6 “in the same circle or in congruent ones”
In the same circle, or in congruent circles:
If two chords are congruent, then their corresponding minor arcs are congruent.
If two minor arcs are congruent, then their corresponding chords are congruent.
inscribed angle
an angle whose vertex is on a circle and whose sides contain chords of the circle
intercepted arc
the arc that lies in the interior of an inscribed angle and has endpoints of the angle
Theorem 11.7 “measure of an inscribed angle”
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
inscribed
if all the vertices of a polygon lie on a circle
circumscried
if all the vertices of a polygon lie on a circle and the circle is circumscribed about the polygon
Theorem 11.8 “inscribed and circumscribed”
If a triangle inscribed in a circle is a right triangle, then the hypotenuse is a diameter of the circle.
If a side of a triangle inscribed in a circle is a diameter of the circle, then the triangle is a right triangle.
Theorem 11.9 “quadrilateral inscribed”
If a quadrilateral can be inscribed in a circle, then its opposite angles of a quadrilateral are supplementary.
If the opposite angles of a quadrilateral are supplementary, then the quadrilateral can be inscribed in a circle
Theorem 11.10 “properties of chords”
If two chords intersect inside a circle, then the measure of each angel formed is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
Theorem 11.11 “intersecting chords”
If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord
standard equation of a circle
In the coordinate plane, the standard equation of a circle with center a (h,k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
rotation
a transformation in which a figure is turned about a fixed point. the fixed point is the center of rotation
angle of rotation
Rays drawn from the center of rotation to a point and its image form an angle
Rotational symmetry
if a figure in a plane can be mapped onto itself by a rotation of 180` or less
