unit 9: circles; theorems only

Question 1

arc addition postulate

Answer 1

the measure of an arc formed by 2 adjacent arcs is the sum of the measures of both arcs

Question 2

congruent central angles theorem

Answer 2

in the same/congruent circle(s), 2 minor arcs are congruent if and only if their central angles are congruent

Question 3

basic equations of circles

Answer 3

1. A= π r^2
2. c = π d
3. arc length= (degree of arc x c) / 360

Question 4

perpendicular tangent theorem

Answer 4

if a line is tangent to a circle, then it is perpendicular to the radius at the point of tangency

Question 5

perpendicular tangent theorem converse

Answer 5

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

Question 6

external tangents congruence theorem

Answer 6

if 2 tangents to a circle share a common point outside the circle, the 2 segments are congruent

Question 7

central angle theorem

Answer 7

in the same/congruent circle(s),
1. congruent arcs have congruent chords
2. congruent chords have congruent arcs

Question 8

perpendicular chord bisector theorem

Answer 8

a diameter that is perpendicular to a chord bisects the chord & its arc

Question 9

equidistant chords theorem

Answer 9

in the same/congruent circle(s),
1. chords equally distant from the center(s) are congruent
2. congruent chords are equally distant from the center(s)

Question 10

inscribed angle theorem

Answer 10

the measure of an inscribed angle is equal to half the measure of its intercepted arc

Question 11

inscribed angle theorem corollaries

Answer 11

1. if 2 inscribed angles intercept the same arc, the angles are congruent
2. an angle inscribed in a semicircle is a right angle
3. if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary

Question 12

tangent & intersected chord theorem

Answer 12

the measure of an angle formed by a chord and a tangent is half the measure of the intercepted arc

Question 13

angle inside the circle theorem

Answer 13

the measure of an angle formed by the intersection of 2 chords in a circle is equal to 1/2 the sum of the intercepted arcs

Question 14

angle outside the circle theorem

Answer 14

the measure of an angle formed by 2 secants, 2 tangents, or a secant and a tangent from a point outside the circle is equal to 1/2 the difference of the intercepted arcs

Question 15

segments of chords theorem

Answer 15

when 2 chords intersect inside a circle, the products of 1 chord’s segments equal the products of the other chord’s segments

rs=tu, r/t=u/s

Question 16

segments of secants theorem

Answer 16

when 2 secants meet at an external point, the product of 1 secant and its external segment equals the product of the other secants and its external segment

rs=tu, r/t=u/s

Question 17

segments of a tangent and a secant theorem

Answer 17

when a secant & tangent meet at an external point, the product of the secant segment and its external segment equal the tangent segment squared

rs=t^2, r/t=t/s