lesson 2.4 curves and circles pp. 54-59

Lesson 2.4 Curves and Circles

pp. 54-59

Lesson 2.4 Curves and Circles

pp. 54-59

Objectives:1. To define a triangle and related

terms.2. To classify curves.3. To define a circle and related terms.

4. To state the Jordan Curve Theorem.

Objectives:1. To define a triangle and related

terms.2. To classify curves.3. To define a circle and related terms.

4. To state the Jordan Curve Theorem.

A triangle is the union of segments that connect three noncollinear points. A triangle is designated by the symbol followed by the three noncollinear points.

A triangle is the union of segments that connect three noncollinear points. A triangle is designated by the symbol followed by the three noncollinear points.

DefinitionDefinitionDefinitionDefinition

TriangleTriangleRR

SS

TT

Denoted: Denoted: RSTRST

TriangleTriangle

opposite sidesopposite sides

RR

SS

TTRTRT

RSRS

STST

A closed curve is a curve that begins and ends at the same point.

A closed curve is a curve that begins and ends at the same point.


A simple curve is a curve that does not intersect itself (unless the starting and ending points coincide).

A simple curve is a curve that does not intersect itself (unless the starting and ending points coincide).


A simple closed curve is a simple curve that is also a closed curve.

A simple closed curve is a simple curve that is also a closed curve.


A circle is the set of all points that are a given distance from a given point in a given plane.

The center of the circle is the given point in the plane.

A circle is the set of all points that are a given distance from a given point in a given plane.

The center of the circle is the given point in the plane.


A radius of a circle is a segment that connects a point on the circle with the center. (The plural of radius is radii.)

A chord of a circle is a segment having both endpoints on the circle.

A radius of a circle is a segment that connects a point on the circle with the center. (The plural of radius is radii.)

A chord of a circle is a segment having both endpoints on the circle.


OO

AA

OO

CCBB

A diameter is a chord that passes through the center of the circle.

An arc is a curve that is a subset of a circle. (symbol: )

A diameter is a chord that passes through the center of the circle.

An arc is a curve that is a subset of a circle. (symbol: )


OO

DD

AACCBB

EE

AEAE

The interior of a circle is the set of all planar points whose distance from the center of the circle is less than the length of the radius (r).

The interior of a circle is the set of all planar points whose distance from the center of the circle is less than the length of the radius (r).


The exterior of a circle is the set of all planar points whose distance from the center of the circle is greater than the length of the radius (r).

The exterior of a circle is the set of all planar points whose distance from the center of the circle is greater than the length of the radius (r).


Theorem 2.1Jordan Curve Theorem. Any simple closed curve divides a plane into three disjoint sets: the curve itself, its interior, and its exterior.

Theorem 2.1Jordan Curve Theorem. Any simple closed curve divides a plane into three disjoint sets: the curve itself, its interior, and its exterior.

A region is the union of a simple closed curve and its interior. The curve is the boundary of the region.

A region is the union of a simple closed curve and its interior. The curve is the boundary of the region.


Homeworkpp. 58-59

Homeworkpp. 58-59

►►A. ExercisesA. ExercisesClassify each figure as (1) a curve, (2) a Classify each figure as (1) a curve, (2) a closed curve, (3) a simple curve, (4) a closed curve, (3) a simple curve, (4) a simple closed curve, (5) or not a curve. simple closed curve, (5) or not a curve. Use the most specific term possible.Use the most specific term possible.11.11.

►►B. ExercisesB. ExercisesUse the figure for exercises 18-22.Use the figure for exercises 18-22.

AA

BB CC DD EE19.19. Name all the angles.Name all the angles.

►►B. ExercisesB. ExercisesUse the figure for exercises 18-22.Use the figure for exercises 18-22.

AA

BB CC DD EE21.21. ABD ABD ADEADE

►►B. ExercisesB. Exercises23.23. If X, Y, and Z are noncollinear, find If X, Y, and Z are noncollinear, find

XY XY YZ YZ XZ. XZ.

►►B. ExercisesB. Exercises Use the figure shown for exercises 25-29.Use the figure shown for exercises 25-29. S S is is the region bounded by rectangle the region bounded by rectangle AHFCAHFC. Tell . Tell whether the statements is true or false.whether the statements is true or false.

CCAA

HH FF

BB DD

GG EE

II

25.25. BCIBCI S = S = BCFBCF

CCAA

HH FF

BB DD

GG EE

II

27.27. SS BGED = BGFCBGED = BGFC


CCAA

HH FF

BB DD

GG EE

II

29.29. ABGHABGH BGFC = ACFH BGFC = ACFH BGBG


■ Cumulative ReviewTrue/False

32. The intersection of two planes can be a single point.


32. The intersection of two planes can be a single point.


33. The intersection of two opposite half-planes is their common edge.


33. The intersection of two opposite half-planes is their common edge.


34. A segment is a curve.


34. A segment is a curve.


35. The Line Separation Postulate asserts that a line separates a plane into three disjoint sets.


35. The Line Separation Postulate asserts that a line separates a plane into three disjoint sets.


36. If planes s and t are parallel, then every line in plane s is parallel to every line in plane t.


36. If planes s and t are parallel, then every line in plane s is parallel to every line in plane t.

Analytic Geometry

Graphing Lines and Curves

Analytic Geometry

Graphing Lines and Curves

Graph y = x + 2Graph y = x + 2

yyxx

Graph y = -x2Graph y = -x2

yyxx

yyxx

Graph y = xGraph y = x

yyxx

1. Graph y = x - 51. Graph y = x - 5

00 -5-5

11 -4-4

22 -3-3

55 00

-1-1 -3-3

yyxx

2. Graph y = 3x2. Graph y = 3x

00 00

11 33

yyxx

3. Graph y = x2 + 13. Graph y = x2 + 1

-1-1 22

00 11

11 22

22 55

-2-2 55

yyxx

4. Graph y = 2x + 34. Graph y = 2x + 3

lesson 2.4 curves and circles pp. 54-59

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