Lesson 2.4 Curves and Circles pp. 54-59
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Transcript of Lesson 2.4 Curves and Circles pp. 54-59
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Lesson 2.4 Curves and Circles
pp. 54-59
Lesson 2.4 Curves and Circles
pp. 54-59
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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.
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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
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TriangleTriangleRR
SS
TT
Denoted: Denoted: RSTRST
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TriangleTriangle
opposite sidesopposite sides
RR
SS
TTRTRT
RSRS
STST
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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.
DefinitionDefinitionDefinitionDefinition
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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).
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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.
DefinitionDefinitionDefinitionDefinition
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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.
DefinitionDefinitionDefinitionDefinition
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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.
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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: )
DefinitionDefinitionDefinitionDefinition
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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).
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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).
DefinitionDefinitionDefinitionDefinition
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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.
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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.
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Homeworkpp. 58-59
Homeworkpp. 58-59
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►►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.
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►►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.13.13.
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►►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.15.15.
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►►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.17.17.
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►►B. ExercisesB. ExercisesUse the figure for exercises 18-22.Use the figure for exercises 18-22.
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BB CC DD EE19.19. Name all the angles.Name all the angles.
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►►B. ExercisesB. ExercisesUse the figure for exercises 18-22.Use the figure for exercises 18-22.
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BB CC DD EE21.21. ABD ABD ADEADE
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►►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.
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►►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.
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HH FF
BB DD
GG EE
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25.25. BCIBCI S = S = BCFBCF
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CCAA
HH FF
BB DD
GG EE
II
27.27. SS BGED = BGFCBGED = BGFC
►►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.
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CCAA
HH FF
BB DD
GG EE
II
29.29. ABGHABGH BGFC = ACFH BGFC = ACFH BGBG
►►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.
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■ Cumulative ReviewTrue/False
32. The intersection of two planes can be a single point.
■ Cumulative ReviewTrue/False
32. The intersection of two planes can be a single point.
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■ Cumulative ReviewTrue/False
33. The intersection of two opposite half-planes is their common edge.
■ Cumulative ReviewTrue/False
33. The intersection of two opposite half-planes is their common edge.
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■ Cumulative ReviewTrue/False
34. A segment is a curve.
■ Cumulative ReviewTrue/False
34. A segment is a curve.
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■ Cumulative ReviewTrue/False
35. The Line Separation Postulate asserts that a line separates a plane into three disjoint sets.
■ Cumulative ReviewTrue/False
35. The Line Separation Postulate asserts that a line separates a plane into three disjoint sets.
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■ Cumulative ReviewTrue/False
36. If planes s and t are parallel, then every line in plane s is parallel to every line in plane t.
■ Cumulative ReviewTrue/False
36. If planes s and t are parallel, then every line in plane s is parallel to every line in plane t.
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Analytic Geometry
Graphing Lines and Curves
Analytic Geometry
Graphing Lines and Curves
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Graph y = x + 2Graph y = x + 2
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Graph y = -x2Graph y = -x2
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Graph y = xGraph y = x
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1. Graph y = x - 51. Graph y = x - 5
00 -5-5
11 -4-4
22 -3-3
55 00
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-1-1 -3-3
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2. Graph y = 3x2. Graph y = 3x
00 00
11 33
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3. Graph y = x2 + 13. Graph y = x2 + 1
-1-1 22
00 11
11 22
22 55
-2-2 55
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4. Graph y = 2x + 34. Graph y = 2x + 3