42 terms

Modern School Mathematics Geometry Chapter 10

Circles
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Tangent
a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point
Point of Tangency
the point where a circle and a tangent intersect
Theorem 10-1
A line in the plane of a circle and containing an interior point of the circle intersects the circle in two points.
Theorem 10-2
In a circle, tangent perpendicular to the radius
Theorem 10-3
In a circle, tangent perpendicular to radius
Common tangent between two circles
is tangent to both
Common internal tangent
crosses through space in the middle
Common external tangent
does not cross through space in the middle
Statements that can be used
Radii of a circle are congruent
In a circle, radii congruent to radii of congruent circles are congruent
Circle
The set of all points equally distant from a given point
Central Angle
An angle in a circle with the vertex on the center
Minor Arc
An arc of a circle whose measure is less than 180 degrees (less then a semicircle)
Major Arc
part of a circle that measures from 180 to 360 (greater than a semicircle)
Semicircle
Half a circle, the arc created by the endpoints of the diameter
Measure of minor arc
the measure of its central angle
Measure of major arc
the difference between 360 and the measure of its associated minor arc
Measure of a semicircle
180 degrees
Congruent Arcs
In congruent circles, have the same measure
Arc Addition Theorem
If the intersection of arcs DE and EF of a circle is the single point E, then mDE+mEF=mDEF
Theorem 10-4
In same or congruent circles, congruent central angles yield congruent arcs
Theorem 10-5
In same or congruent circles, congruent arcs yield congruent central angles
Chords
a segment that both endpoints on the circle
Arc of a chord
The part of the circle cut off by those endpoints
Theorem 10-6
In same or congruent circles, congruent chords yield congruent arcs
Theorem 10-7
In same or congruent circles, congruent arcs yield congruent chords
Theorem 10-8
Diameters perpendicular to a chord yield bisected chord and arcs
Theorem 10-9
In same or congruent circles, chords equidistant from center arcs are congruent
Theorem 10-10
In same or congruent circles, congruent chords are equidistant from the center
Inscribed angle
An angle in the circle whose vertex is on the circle, and both sides (rays) intersect the circle
An angle intercepts and arc if
1. The vertex is on the circle
2. Both rays intersect the circle
Theorem 10-11
m(inscribed ange)= 1/2 intercepted arc
Corollary 1
An angle inscribed in a semicircle is a right angle
Corollary 2
If a quadrilateral is inscribed in a circle, that yields supplementary opposite angles.
Corollary 3
In a circle, congruent ars yield congruent inscribed angles
Secant
a line that intersects a circle at two points. A line that contains a chord.
Theorem 10-13
In a circle, m(interior angle) = the average of the intercepted arcs or:
1/2(∑ or intercepted arcs)
Theorem 10-14
In a circle, m(exterior ∠) = ½ the difference of the intercepted arcs, or:
In a circle, m(exterior ∠) = ½(arc₂-arc₁)
Secant/Secant
Theorem 10-15
tan/secant= ½(arc₂-arc₁)
Theorem 10-16
If the chords bisect inside the circle, product of 2 part of one chord=the product of 2 parts of other chord (ab=xy)
Theorem 10-17
If secants intersect out side the circle:
(whole part₁)(outside₁)=(whole part₂)(outside₂)
Theorem 10-18
tan²= (whole part)(outside)
Important!
From a point outside a circle, both tangents are congruent.