Modern School Mathematics Geometry Chapter 10

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Circles

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.

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