Transcript G P EOMETRY ROJECT

GEOMETRY PROJECT
Meagan Farber
CHAPTER ONE
Basics of Geometry
1.1)
PATTERNS AND INDUCTIVE REASONING

Terms to know
Conjecture- is an unproven statements that is based
one observations
 Inductive reasoning- looking for patterns and making
conjectures
 Counterexample- an example that shows a conjecture
is false

1.2)
POINTS, LINES, AND PLANES
Point has no dimension. Usually represented by a
small dot
 Line extends in one dimension. Usually
represented by a straight line with two arrow
heads to indicate that the line extends.
 Plane extends in two dimensions. It usually
represented by a shape that looks like a tabletop
 Collinear Points lie on the same line
 Coplanar Points life on the same plane
 Two or more figures intersect if they have one or
more points in common
 The intersection of the figures is the set of points
the figures have in common

Line:
A
Point
B
Line Segment:
A
Line
B
Ray:
Plane:
A
a
B
b
Opposite Rays:
A
C
B
d
c
1.3)
SEGMENTS AND THEIR MEASURES
Postulate 1: Rulers
Postulate

The points on a line can be
matched one to one with the
real numbers. The real
number that corresponds to
a point is the coordinate of
the point. The distance
between points A and B,
written as AB, is the
absolute value of the
difference between the
coordinates of A and B. AB
is also called the length of
AB


Postulates/axioms- rules
that are accepted without
proof
Theorems- rules that are
proved
Names of points
A
x1
B
x2
Coordinates of points
AB= |x2 – x1|
Postulate 2: Segment
Addition Postulate

If B is between A and
C, then AB + BC =
AC. If AB + BC = AC,
then B is between A
and C
The Distance Formula

If A(x1,y1) and B(x2,y2)
are points in a
coordinate plane, then
the distance between A
and B is

A
B

C
AB= √(x2-x1)^2 + (y2y1)^2
Congruent Segmentssegments that have the
same length
Distance Formula

(AB)^2 = (x2-x1)^2 +
(y2-y1)^2
Pythagorean Theorem
1.4)
ANGLES AND THEIR MEASURES
An angle is two different rays that have the
same initial point. The rays are the sides of the
angle. The initial point is the vertex of the angle.
Angles that have the same measure are
congruent angles
 Postulate 3: Protractor Postulate


Consider a point A on one side of BC. The rays of the
form BA can be matched one to one with real
numbers from 0 to 180
Postulate 4: Angle Addition Postulate
Classifying Angles
1.5)
SEGMENT AND ANGLE BISECTORS
Midpoint- point that bisects the segments into
two congruent segments.
 Segment bisector- segment, ray, line, or plane
that intersects a segment at its midpoint

Midpoint Formula Angle bisector- is a ray that divides an angle into
two adjacent that are congruent

1.6)
ANGLE PAIR RELATIONSHIPS
Vertical angles- if sides form two pairs of opposite rays
 Linear pair- if their non-common sides are opposite
rays
 <1 and <3 are vertical angles


<1 and <2 are a linear pair
Complementary
Adjacent
Supplementary non adjacent
Complementary non adjacent
1.7)
INTRO TO PERIMETER, CIRCUMFERENCE, AND
AREA

Square- side length s



Triangle- side length a, b, and c, base b and
height h



P = 4s
A= s^2
P= a + b + c
A= ½bh
Rectangle- length l and width w
P= 2l + 2w
 A=lw


Circle= radius r
C= 2 r
 A= r^2

CHAPTER TWO
Reasoning and Proof
2.1)
CONDITIONAL STATEMENTS
Conditional statement- has two parts a
hypothesis and a conclusion
 converse- formed by switching the hypothesis and
conclusion
 negation- writing the negative of the statement
 inverse-when you negate the hypothesis and
conclusion
 Contra positive-when you negate the hypothesis
and conclusion of the converse
 Equivalent statements- two statements are both
true or both false


Point, line and Plane Postulates







Postulate 5: through any two points there exists
exactly one line
Postulate 6: a line contains at least two points
Postulate 7:if two lines intersect, then their
intersection is exactly one point
Postulate 8: through any 3 non collinear points there
exists exactly one point
Postulate 9: a plane contains at least 3 non collinear
points
Postulate 10: if two points lie in a plane, then the line
containing them lies in the plane
Postulate 11: if two planes intersect, then their
intersection is a line
2.2)DEFINITIONS AND BICONDITIONAL STATEMENTS

Perpendicular lines- if they intersect to form a right angle

Biconditional statement- a statement that contains “ if and only if”
2.3)DEDUCTIVE REASONING
 Logical argument- deductive reasoning uses facts, definitions, and accepted
properties in a logical order


Law of Detachment- if p→q is a true conditional statement and p is
true, then q is true
Law of Syllogism- p→q and q→r are true conditional statements then
p→r is true
2.4)REASONING WITH PROPERTIES FROM ALGEBRA








Addition property- if a=b, then a+c=b+c
Subtraction property- if a=b, then a-c=b-c
Multiplication property- if a=b, then ac=bc
Division property- if a=b and c≠0, then a/c=b/c
Reflexive property-for any real number a, a=a
Symmetric property- if a=b, then b=a
Transitive property- if a=b and b=c, then a=c
Substitution property= if a=b, then a can be substituted for b in any
equation or expression
2.5)PROVING STATEMENTS

ABOUT SEGMENTS
Theorem 2.1- Properties of Segment Congruence

Segment congruence is reflexive, symmetric and transitive
 Reflexive- for any segment AB, AB≈AB
 Symmetric- if AB≈CD, then CD≈AB
 transitive-if AB≈CD, and CD≈EF, then AB≈EF
2.6)PROVING STATEMENTS ABOUT ANGLES

Theorem 2.2- Properties of Angles Congruence

Angle congruence is reflexive, symmetric and transitive

Reflexive- for any angle A, <A≈<A

Symmetric- If <A≈<B, then <B ≈<A

Transitive- if <A≈<B and <B≈<C, then <A≈<C
2.3) RIGHT ANGLE CONGRUENCE THEOREM

All right angles are congruent
2.4) CONGRUENT

If two angles are supplementary to the same angle they are congruent
2.5) CONGRUENT

SUPPLEMENTS THEOREM
COMPLEMENTS THEOREM
If two angles are complementary to the same angle then the two angles are congruent
POSTULATE 12) LINEAR PAIR POSTULATE

If two angles form a linear pair, then they are supplementary
THEOREM 2.6) VERTICAL ANGLES THEOREM

Vertical angles are congruent
CHAPTER THREE
Perpendicular and Parallel Lines
3.1) LINES AND ANGLES

Parallel lines- two lines that are coplanar and do not intersect

Skew lines- lines that do not intersect and are not coplanar

Parallel planes- two planes that do not intersect



Postulate 13) parallel postulate
 If there is a line and a point not on the line, then there is exactly one
line through the point parallel to the given line
Postulate 14) perpendicular postulate
 If there is a line and a point not on the line, then there is exactly one
line through the point perpendicular to the given line
Transversal- a line that intersects two or more coplanar lines at different
points
Corresponding angles = 1 and 5
Alternate exterior angles = 1 and 5
Alternate interior angles= 3 and 6
Consecutive interior angle= 3 and 5
3.2) PROOF AND PERPENDICULAR LINES



Theorem 3.1- if two lines intersect to form a linear pair of
congruent angles, then the lines are perpendicular
Theorem 3.2- if two sides of two adjacent acute angles are
perpendicular, then the angles are complementary
Theorem 3.3- if two lines are perpendicular, then they intersect to
form four right angles
3.3)PARALLEL LINES AND TRANSVERSALS

Postulate 15) Corresponding Angles Postulates





If two parallel lines are cut by a transversal, then the pairs of
corresponding angles are congruent
Theorem 3.4- if two parallel lines are cut by a transversal, then the pairs
of alternate interior angles are congruent
Theorem 3.5- if two parallel lines are cut by a transversal, then the pair of
consecutive interior angles are supplementary
Theorem 3.6- if two parallel lines are cut by transversal, then the pairs of
alternate exterior angles are congruent
Theorem 3.7- if a transversal is perpendicular to one of two parallel lines,
then its is perpendicular to the other
3.4 PROVING LINES ARE PARALLEL




Postulate 16) Corresponding angles converse
 If two lines are cut by a transversal so that corresponding angles are congruent,
then the lines are parallel
Theorem 3.8- if two liens are cut by a transversal so that alternate interior angles
are congruent then the lines are parallel
Theorem 3.9- if two lines are cut by a transversal so that consecutive interior angles
are supplementary, then the lines are parallel
Theorem 3.10- if two lines are cut by a transversal so that alternate exterior angles
are congruent then the lines are parallel
3.5 USING PROPERTIES OF PARALLEL LINES


Theorem 3.11- if two lines are parallel to the same line then they are parallel to each
other
Theorem 3.12- in a plane, if two lines are perpendicular to the same line, then they
are parallel to each other
3.6 PARALLEL LINES IN THE COORDINATE PLANE

Postulate 17) Slopes of parallel lines

In a coordinate plane, two non vertical lines are parallel if and only if they have
the same slope, any two verticall lines are parallel
3.7 PERPENDICULAR LINES IN THE COORDINATE PLANE

Postulate 18) Slopes of perpendicular lines

In a coordinate plane, two non vertical lines are perpendicular, if and only if
they product of their slopes is -1. vertical and horizontal lines are perpendicular
CHAPTER 4
Congruent Triangles
4.1 TRIANGLES AND ANGLES



Theorem 4.1 Triangle Sum Theorem- the sum of the measures of
the interior angles of a triangle is 180°
Theorem 4.2 Exterior Angle Theorem- the measure of an exterior
angle of a triangle is equal to the sum of the measure of the two
nonadjacent interior angles
Corollary to the triangle sum theorem- the acute angles of a right
triangle are complementary
4.2 CONGRUENCE AND TRIANGLES


Theorem 4.3 Third Angles Theorem- if two angles of one triangle
are congruent to two angles of another triangle, then the third
angles are also congruent
Theorem 4.4 Properties of Congruent Triangles

Reflexive property- every triangle is congruent to itself

Symmetric property- if ∆abc≈∆def, then ∆def≈∆abc

Transitive property- if ∆abc≈∆def and ∆def≈∆jkl, then ∆abc ≈
∆ jkl
4.3 PROVING TRIANGLES ARE CONGRUENT: SSS AND SAS


Postulate 19- Side-side-side congruence postulate
 If there sides of one triangle are congruent to three sides of a second
triangle then the two triangles are congruent
Postulate 20- Side-angle-Side congruence postulate
 If two sides and the included angle of one triangle are congruent to
two sides and the included angle of a second triangle, then the two
triangles are congruent
4.4 PROVING TRIANGLES ARE CONGRUENT: ASA AND AAS


Postulate 21- Angle-side-angle congruence postulate
 If two angles and the included side of one triangle are congruent to
two angles and the included side of a second triangle, then the two
triangles are congruent
Postulate 22- Angle-angle-side congruence postulate
 If two angles and a non included side of one triangle are congruent to
two angles and the corresponding non included side of a second
corresponding non included side of a second triangle, then the two
triangles are congruent
4.6 ISOSCELES, EQUILATERAL, AND RIGHT
TRIANGLES





Theorem 4.6- if two sides of a triangles are congruent, then the
angles opposite them are congruent
Theorem 4.7- if two angles of a triangle are congruent, then the
sides opposite them are congruent
Corollary to theorem 4.6- if a triangle is equilateral, then its is
equiangular
Corollary to theorem 4.7- if a triangle is equiangular, then it is
equilateral
Theorem 4.8- if the hypotenuse and leg of a right triangle are
congruent to the hypotenuse and leg of a second right triangle ,
then the two triangles are congruent
CHAPTER FIVE
Properties of Triangles
5.1 PERPENDICULARS AND BISECTORS
Theorem 5.1- if a point is on the perpendicular
bisector of a segment, then it is equidistant from
the endpoints of the segment
 Theorem 5.2- if a point is equidistant from the
endpoints of a segment, then its on the
perpendicular bisector of the segment
 Theorem 5.3- if a point is on the bisector of an
angle, then it is equidistant from the two sides of
the angle
 Theorem 5.4- if a point is in the interior of an
angle and is equidistant from the sides of the
angle, then it lies on the bisector of an angle

5.2 BISECTORS OF A TRIANGLE


Theorem 5.5- the perpendicular bisectors of a triangle intersect at
a point that is equidistant from the vertices of the triangle
Theorem 5.6- the angle bisectors of a triangle intersect at a point
that is equidistant from the sides of the triangle
5.3 MEDIANS AND ALTITUDES OF A TRIANGLE


Theorem 5.7- the medians of a triangle intersect at a point that is
two thirds of the distance from each vertex to the midpoint of the
opposite sides
Theorem 5.8- the lines containing the altitudes of a triangle are
concurrent
5.4 MIDSEGMENT THEOREM

Theorem 5.9- the segment connecting the midpoints of two sides
of a triangle is parallel to the third sides and is half as long
5.5 INEQUALITIES IN ONE TRIANGLE


Theorem 5.10- if one side of a triangle is longer than another side,
then the angle opposite the longer side is larger than the angle
opposite the shorter side
Theorem 5.11- if one angle of a triangle is larger than another
angle, then the side opposite the larger angle is longer than the
side opposite the smaller angle


Theorem 5.12- the measure of an exterior angle of a triangle is
greater than the measure of either of the two nonadjacent interior
angles
Theorem 5.13- the sum of the lengths of any two sides of a
triangle is greater than the length of the third side
5.6 INDIRECT PROOF AND INEQUALITIES IN TWO
TRIANGLES


Theorem 5.14- if two dies of one triangle are congruent to
two sides of another triangle, and the included angle of the
first is larger than the included angle of the second, then
the third side of the first is longer than the third side of the
second
Theorem 5.15- if two sides of one triangle are congruent to
two sides of another triangle, and the third side of the first
is longer than the third side of the second, then the
included angle of the first is larger than the included angle
of the second
CHAPTER SIX
Quadrilaterals
POLYGONS
Number of
Sides
Type of
Polygon
Number of
Sides
Type of
Polygon
3
1.Triangle
8
6.Octagon
4
2.Quadrilateral 9
7.Nonagon
5
3.Pentagon
10
8.Decagon
6
4.Hexagon
12
9.Dodecagon
7
5.heptagon
N
10.N-gon
2
1
4
3
9
5
6
8
Theorem 6.1- the sum of the measures of the interior angles of a
quadrilateral is 360°
6.2 PROPERTIES OF PARALLELOGRAMS




Theorem 6.2- if a quadrilateral is a parallelogram, then its
opposite sides are congruent
Theorem 6.3- if a quadrilateral is a parallelogram, tehn its
opposites angles are congruent
Theorem 6.4- if a quadrilateral is a parallelogram, tehn its
consecutive angles are supplementary
Theorem 6.5- if a quadrilateral is a parallelogram, then its
diagonals bisect each other
6.3 PROVING QUADRILATERAL ARE PARALLELOGRAMS





Theorem 6.6- if both pairs of opposites sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram
Theorem 6.7- if both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is parallelogram
Theorem 6.8- if an angle of a quadrilateral is supplementary to both of its
consecutive angles, then the quadrilateral is a parallelogram
Theorem 6.9- if the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram
Theorem 6.10- if one pair of opposite sides of a quadrilateral are
congruent and parallel then the quadrilateral is a parallelogram
6.4 RHOMBUSES, RECTANGLES, AND SQUARES

Rhombus is a parallelogram with four congruent sides

A rectangle is a parallelogram with four right angles







A square is a parallelogram with four congruent sides and four
right angles
Rhombus corollary- a quadrilateral is a rhombus if and only if it
has four congruent sides
Rectangle corollary- a quadrilateral is a rectangle if and only if its
has four right angles
Square corollary- a quadrilateral is a square if and only if it’s a
rhombus and a rectangle
Theorem 6.11- a parallelogram is a rhombus if and only if its
diagonals are perpendicular
Theorem 6.12- a parallelogram is a rhombus if and only if each
diagonal bisects a pair of opposite angles
Theorem 6.13- a parallelogram is a rectangle if and only if its
diagonals are congruent
6.5 TRAPEZOIDS AND KITES
Theorem 6.14- if a trapezoid is isosceles, then
each pair of base angles is congruent
 Theorem 6.15- if a trapezoid has a pair of
congruent base angles, then it is an isosceles
trapezoid
 Theorem 6.16- a trapezoid is isosceles if and only
if its diagonals are congruent
 Theorem 6.17- the midsegment of a trapezoid is
parallel to each base and its length is one half the
sum of the lengths of the bases
 Theorem 6.18- if a quadrilateral is a kite, then its
diagonals are perpendicular
 Theorem 6.19- if a quadrilateral is a kite, then
exactly one pair of opposite angles are congruent

CHAPTER SEVEN
Transformations
7.1 RIGID MOTION IN A PLANE




image- new figure
preimage- the original figure
Transformation- the operation that maps or moves the preimage onto the
images
Isometry- a transformation that preserves lengths
7.2 REFLECTIONS

reflection- type of transformation uses a line that acts like a mirror , with
an image reflected in the line

Line of reflection- the mirror line

Line of symmetry- the figure can be mapped onto itself by a reflection line
7.3 ROTATIONS




Rotation- transformation in which a figure is turned about a fixed point
Center of rotation- the fixed point
Angle of rotation- rays drawn from the center of rotation to a point and its
image form an angle
Rotational symmetry- the figure can be mapped onto itself by roation of
180°
7.4 TRANSLATIONS AND VECTORS


Theorem 7.4- a translation is isometry
Theorem 7.5- if lines k and m are parallel, then a reflection
in line k followed by a reflection in line m is a translation.
If P” is the image of p, then the following is true: PP” is
perpendicular to k&m. PP”= 2d, where d is the distance
between k&m.
7.5 GLIDE REFLECTIONS AND COMPOSITIONS



Glide reflection- transformation in which every point p is
mapped onto a point p” by the following steps: a translation
maps p onto p”. A reflection in a line k parallel to the
direction of the translation maps p’ onto p”
Composition- when two or more transformations are
combined to produce a single transformation
Theorem 7.6- the composition of two or more isometries is
an isometry
CHAPTER EIGHT
Similarity
8.1 RATIO AND PROPORTION
Properties of proportions
 Cross products property- the product of the extremes equals
the product of the means
 Reciprocal property- if two ratios are equal, then their
reciprocals are also equal

8.2 PROBLEMS SOLVING IN GEOMETRY WITH PROPORTIONS
Additional properties of proportions


If a/b=c/d then a/c=b/d

If a/b=c/d then a+b/b=c+d/d
8.3 SIMILAR POLYGONS

Theorem 8.1- if two polygons are similar, then the ratio of their
perimeter is equal to the ratios of their corresponding side lengths
8.4 SIMILAR TRIANGLES

Postulate 25- if two angles of one triangle are congruent to two
angles of another triangle, then the two triangles are similar
8.5 PROVING TRIANGLES ARE SIMILAR`
`
Theorem 8.2- if the lengths of the corresponding sides of two triangles are
proportional, then the triangles are similar

Theorem 8.3- if an angle of one triangle is congruent to an angle of a
second triangle and the lengths of the sides including these angles are
proportional then the triangles are similar

8.6 PROPORTIONS AND SIMILAR TRIANGLES




Theorem 8.4- if a line parallel to one side of triangle intersects the other
two sides, the it divides the two sides proportionally
Theorem 8.5- if a line divides two sides of a triangle proportionally then it
is parallel to the third side
Theorem 8.6- if three parallel lines intersect two transversals, then they
divide the transversals proportionally
Theorem 8.7- if a ray bisects an angle of a triangle then it divides the
opposite side into segments whose lengths are proportional to the lengths
of the other two sides
8.7 DILATIONS

Dilation- with center c and scale factor k is a transformatin that maps
ever point p in the plane to point p’ so that the following properties are
true: if p is not the center point c, then the image point p’ lies on CP, the
scale factor k is a postivie number such that k=cp’/cp, and k≠1. if p is the
center point c, then p=p’
CHAPTER NINE
Right triangles and trigonometry
9.1 SIMILAR RIGHT TRIANGLES



Theorem 9.1- if the altitude is drawn to the hypotenuse of a right
triangle., then the two triangles formed are similar to the original
triangle and to each other
Theorem 9.2- in a right triangle, the altitude from the right angle to
the hypotenuse divides the hypotenuse into two segments
Theorem 9.3- in a right triangle the altitude from the right angle to
the hypotenuse divides the hypotenuse into two segmetns. The length
of each leg of the right triangle is the geometric mean of the lengths of
the hypotenuse and the segment of the hypotenuse that is adjacent to
the leg
9.4 SPECIAL RIGH TRIANGLES


Theorem 9.8- in a 45-45-90 degree triangle the hypotenuse is √2
times as long as each leg
Theorem 9.9- in a 30-60-90 degree triangle the hypotenuse is
twice as long as the shorter leg, and the longer leg is √3 times as
long as the shorter leg
9.5 TRIGONOMETRIC RATIOS
9.7 VECTORS

Adding vectors: sum of two vectors- the sum of u=(a1,b1) and
v=(a2,b2) is u+v= (a1+a2, b1+b2)
CHAPTER 10
Circles
10.1 TANGENTS TO CIRCLES



Theorem 10.1- if a line is tangent to a circle, then it is
perpendicular to the radius drawn to the point of tangency
Theorem 10.2- in a plane, if a line is perpendicular to a radius of a
circle at its endpoint on the circle, then hte line is tangent to the
circle
Theorem 10.3- if two segments from the same exterior point are
tangent to a circle, then they are congruent
10.2 ARCS AND CHORDS





Postulate 26- the measure of an arc formed by two adjacent arcs
is the sum of the measures of the two arcs
Theorem 10.4- if the same circle, or in a congruent circles, two
minor arcs are congruent if and only if their corresponding chords
are congruent
Theorem 10.5- if a diameter of a circle is perpendicular to a chord,
then the diameter bisects the chord and its arc
Theorem 10.6- if one chord is a perpendicular bisector of an other
chord, then the first chord is the diameter
Theorem 10.7- in the same circle, or in two congruent circles, two
chords are congruent if and only if they are equidistant from the
center
10.3 INSCRIBED ANGLES







Theorem 10.8- if an angle is inscribed in a circle, then its measure is
half the measure of its intercepted arc
Theorem 10.9- if two angles of a circle intercept the same arc, then the
angles are congruent
Theorem 10.10- if a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle
Theorem 10.11- a quadrilateral can be inscribed in a circle if and only
if opposites = 180°
Theorem 10.12- if a tangent and a chord intersect at a point on a circle
the measure of each angle formed is one half the measure of its
intercepted arc
Theorem 10.13- if two chords intersect in the interior of a circle, then
the measure of each angle is one half the sum of the measures of the
arcs intercepted by the angle and its vertical angle
Theorem 10.14- if a tangent and a secant, two tangents, or two secants
intersect in the exterior of a circle, then the measures of the angle
formed is one half the difference of the measure of the intercepted arcs
10.5 SEGMENT LENGTHS IN CIRCLES



Theorem 10.15- if two chords intersect in the interior of 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
Theorem 10.16- if two secant segments share the same
endpoint outside a circle, then the product of the length of
one secant segment and the length of its external segment
equals the product of the length of the other secant
segment and the length of its external segment
Theorem 10.17- if a secant segment and a tangent segment
share an endpoint outside a circle, then the product of the
length of the secant segment and the length of its external
segment equals the square of the length of the tangent
segment
CHAPTER 11
Area of polygons and circles
11.1 ANGLE MEASURES IN POLYGONS
Theorem 11.1- the sum of the measures of the
interior angles of a convex n-gon is (n-2)*180°
 Corollary to theorem 11.1- the measure of each
interior angle of a regular n-gon is
 Theorem 11.2- the sum of the measure of the
exterior angles of a convex polygon, one angle at
each vertex is 360
 Corollary to theorem 11.2- the measure of each
exterior angle of a regular n-gon is 360°/n

11.2 AREAS OF REGULAR POLYGONS


Theorem 11.3- the area of an equilateral triangle is one fourth the
square of the length of the side time √3
Theorem 11.4- the area of a regular n-gon with side length s is
half the product of the apothem a and the perimeter p, so A= ½ap
or A=12a*ns
11.3 PERIMETERS AND AREAS OF SIMILAR FIGURES

Theorem 11.5- if two polygons are similar with the lengths of
corresponding sides in the ratio a:b, then the areas will be
a^2:b^2
11.4 CIRCUMFERENCE AND ARC LENGTH


Theorem 11.6- the circumference c of a circle is c= d or c=2 r,
where d is diameter and r is the radius
Arc length corollary- in a circle the ratio of the length of a given
arc to circumference is equal to the ratio of the measure of the arc
to 360°
11.5 AREAS OF CIRCLES AND SECTORS


Theorem 11.7- the areas of a circle is times the square of the
radius. A= r^2
Theorem11.8- the ratio to the area a of a sector of a circle to the
area of the circle is equal to the ratio of the measure of the
intercepted arc to 360°
11.6 GEOMETRIC PROBABILITY

Probability and length- let AB be segment that contains the
segment CD. If a point K on AB is chosen at random, then the
probability that it is on CD is as follows: P(point k is on CD) =
length of CD/ length of AB