Geometry 1 Unit 1: Basics of Geometry

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Geometry 1 Unit 1 Basics of Geometry
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Geometry 1 Unit 1

• 1.1 Patterns and Inductive Reasoning

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EXAMPLE 1
Describe a visual pattern
SOLUTION
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for Examples 1 and 2
GUIDED PRACTICE
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EXAMPLE 2
Describe a number pattern
Describe the pattern in the numbers 7, 21, 63,
189, and write the next three numbers in
the pattern.
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for Examples 1 and 2
GUIDED PRACTICE
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Patterns and Inductive Reasoning

• Conjecture
• An unproven statement that is based on
  observations.
• Inductive Reasoning
• The process of looking for patterns and making
  conjectures.

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EXAMPLE 3
Make a conjecture
Given five students, make a conjecture about the
number of different handshakes that can take
place.
SOLUTION
Make a table and look for a pattern. Notice the
pattern in how the number of connections
increases. You can use the pattern to make a
conjecture.
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EXAMPLE 3
Make a conjecture
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EXAMPLE 4
Make and test a conjecture
Numbers such as 3, 4, and 5 are called
consecutive integers. Make and test a conjecture
about the sum of any three consecutive integers.
SOLUTION
STEP 1
Find a pattern using a few groups of small
numbers.
3 4 5
7 8 9
12
12
10 11 12
16 17 18
33
51
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EXAMPLE 4
Make and test a conjecture
STEP 1
Test your conjecture using other numbers. For
example, test that it works with the groups 1,
0, 1 and 100, 101, 102.
100 101 102
303
1 0 1
0
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for Examples 3 and 4
GUIDED PRACTICE
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Patterns and Inductive Reasoning

• Counterexample
• An example that shows a conjecture is false.
• All Math teachers are male.
• Mrs. Beery, Ms. Wildermuth, Mrs. Hodge, Mrs.
  Cherry, Mrs. Frimer, Mrs. Dolezal are all
  counterexamples.

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EXAMPLE 5
Find a counterexample
A student makes the following conjecture about
the sum of two numbers. Find a counterexample to
disprove the students conjecture.
Conjecture The sum of two numbers is always
greater than the larger number.
SOLUTION
To find a counterexample, you need to find a sum
that is less than the larger number.
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EXAMPLE 5
Find a counterexample
2 3
5
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for Examples 5 and 6
GUIDED PRACTICE
Conjecture The value of x2 is always greater
than the value of x.
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Unit 1-Basics of Geometry

• 1.2 Points, Lines and Planes

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Points, Lines, and Planes

• Definition
• Uses known words to describe a new word.
• Undefined terms
• Words that lack a formal definition.
• In Geometry it is important to have a general
  agreement about these words.
• The building blocks of Geometry are undefined
  terms.

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Points, Lines, and Planes

• The 3 Building Blocks of Geometry
• Point
• Line
• Plane
• These are called the building blocks of
  geometry because these terms lay the foundation
  for Geometry.
•  

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Points, Lines, and Planes

• Point
• The most basic building block of Geometry
• Has no size
• A location in space
• Represented with a dot
• Named with a Capital Letter
•  

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Points, Lines, and Planes
Example point P   P
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Points, Lines, and Planes

• Line
• Set of infinitely many points
• One dimensional, has no thickness
• Goes on forever in both directions
• Named using any two points on the line with the
  line symbol over them, or a lowercase script
  letter

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Points, Lines, and Planes
Example line AB, AB, BA or l B A   2
points determine a line
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Points, Lines, and Planes

• Plane
• Has length and width, but no thickness
• A flat surface that extends infinitely in
  2-dimensions (length and width)
• Represented with a four-sided figure like a
  tilted piece of paper, drawn in perspective
• Named with a script capital letter or 3 points in
  the plane

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Points, Lines, and Planes
Example Plane P or plane ABC A C B
P 3 noncollinear points determine a plane
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Points, Lines, and Planes

• Collinear
• Points that lie on the same line
• Points A, B, and C are Collinear

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Points, Lines, and Planes

• Coplanar
• Points that lie on the same plane
• Points D, E, and F are Coplanar

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Points, Lines, and Planes

• Line Segment
• Two points (called the endpoints) and all the
  points between them that are collinear with those
  two points
• Named line segment AB, AB, or BA
• line AB segment AB 
• A B A B

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Points, Lines, and Planes

• Ray
• Part of a line that starts at a point and extends
  infinitely in one direction.
• Initial Point
• Starting point for a ray.
• Ray CD, or CD, is part of CD that contains point
  C and all points on line CD that are on the same
  side as of C as D
• It begins at C and goes through D and on
  forever

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Segments and Their Measures

• Between
• When three points are collinear, you can say that
  one point is between the other two.

• Point B is between A and C

• Point E is NOT between D and F

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Points, Lines, and Planes

• Opposite Rays
• If C is between A and B, then CA and CB are
  opposite rays.
• Together they make a line.

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Points, Lines, and Planes

• C Y D C Y D C Y D
•  
• Line CD Ray DC Ray CD
•  
• CD and CY represent the same ray.
•  Notice CD is not the same as DC.
• ray CD is not opposite to ray DC

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Points, Lines, and Planes

• The intersection of two lines is a point.
• The intersection of two planes is a line.

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Unit 1-Basics of Geometry

• 1.3 Segments and Their Measures

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Segments and Their Measures

• Postulates
• Rules that are accepted without proof.
• Also called axioms

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Segments and Their Measures

• Ruler 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
  called 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.

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Segments and Their Measures

• Segment length can be given in several different
  ways. The following all mean the same thing.
• A to B equals 2 inches
• AB 2 in.
• mAB 2 inches

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Segments and Their Measures

• Example 1
• Measure the length of the segment to the nearest
  millimeter.

D
E
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Segments and Their Measures

• Between
• When three points are collinear, you can say that
  one point is between the other two.

• Point B is between A and C

• Point E is NOT between D and F

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Segments and Their Measures

• 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.

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Segments and Their Measures

• Example 2
• Two friends leave their homes and walk in a
  straight line toward the others home. When they
  meet, one has walked 425 yards and the other has
  walked 267 yards. How far apart are their homes?

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Segments and Their Measures

• The Distance Formula
• A formula for computing the distance between two
  points in a coordinate plane.
• If A(x1,y1) and B(x2,y2) are points in a
  coordinate plane, then the distance between A and
  B is

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Segments and Their Measures

• Example 3
• Find the lengths of the segments. Tell whether
  any of the segments have the same length.

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Segments and Their Measures

• Congruent
• Two segments are congruent if and only if they
  have the same measure.
• The symbol for congruence is ?.
• We use between equal numbers and ? between
  congruent figures.

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Segments and Their Measures
Markings on figures are used to show congruence.
Use identical markings for each pair of congruent
parts.   A 2.5 B AB DC 2.5 AB ?
DC D 2.5 C AD ? BC
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Segments and Their Measures

• Distance Formula and Pythagorean Theorem

(AB)2 (x2 x1)2 (y2 y1)2
c2 a2 b2
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Segments and Their Measures

• Example 4
• On the map, the city blocks are 410 feet apart
  east-west and 370 feet apart north south.
• Find the walking distance between C and D.
• What would the distance be if a diagonal street
  existed between the two points?

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Unit 1-Basics of Geometry

• 1.4 Angles and Their Measures

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Angles and Their Measures

• Angle
• Formed by two rays that share a common endpoint.
• Sides
• The rays that make the angle.
• Vertex
• The initial point of the rays.

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Angles and Their Measures

• When naming an angle, the vertex must be the
  middle letter.
• angle CAT, angle TAC, ?CAT or ?TAC

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Angles and Their Measures

• If a vertex has only one angle then you can name
  it with that letter alone.
• ?TAC could also be called ?A.

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Angles and Their Measures

• Example 1
• Name all the angles in the following drawing
• 
  

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Angles and Their Measures

• Protractor
• Geometry tool used to measure angles. Angles are
  measured in Degrees.
• Things to know
• A full circle is 360 degrees, or 360º.
• A line is 180º.

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Angles and Their Measures

• Measure of an Angle
• The smallest rotation between the two sides of
  the angle.
• Congruent angles
• Angles that have the same measure.

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Angles and Their Measures

• Angle measure notation
• Use an m before the angle symbol to show the
  measure
• m?A 34º or measure of ?A 34º

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Angles and Their Measures

• Protractor Postulate
• Consider a point A not on OB. The rays of the
  form OA can be matched one to one with the real
  numbers from 0 to 180.
• The measure of an angle is equal to the number on
  the protractor which one side of the angle passes
  through when the other side goes through the
  number zero on the same scale.

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Angles and Their Measures

• Step 1 Place the center mark of the protractor
  on the vertex.
• Step 2 Line up the 0-mark with one side of the
  angle.
• Step 3 Read the measure on the protractor scale.
• Be sure you are reading the scale with the
  0-mark you are using.

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Angles and Their Measures

• Interior
• A point is in the interior if it is between
  points that lie on each side of the angle.
• Exterior
• A point is in the exterior of an angle if it is
  not on the angle or in its interior.

E
exterior
D
interior
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Angles and Their Measures

• Angle Addition Postulate
• If P is in the interior of ?RST, then
• m?RSP m?PST m?RST

R
m ?RST
m ?RSP
S
P
m ?PST
T
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Angles and Their Measures

• Example 2
• The backyard of a house is illuminated by a light
  fixture that has two bulbs.
• Each bulb illuminates an angle of 120.
• If the angle illuminated only by the right bulb
  is 35, what is the angle illuminated by both
  bulbs?

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Angles and Their Measures

• Acute Angle
• An angle whose measure is greater than 0 and
  less than 90º.

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Angles and Their Measures

• Right Angle
• An angle whose measure is 90º

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Angles and Their Measures

• Obtuse Angle
• An angle whose measure is greater than 90º and
  less than 180º.

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Angles and Their Measures

• Straight Angle
• An angle whose measure is 180.

A
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Angles and Their Measures

• Example 3
• Plot the following points.
• A(-3, -1), B(-1, 1), C(2, 4), D(2, 1), and E(2,
  -2)
• Measure and classify the following angles as
  acute, right, obtuse or straight.
• a. ?DBE
• b. ?EBC
• c. ?ABC
• d. ?ABD

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Angles and Their Measures
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Angles and Their Measures

• Adjacent Angles
• Angles that share a common vertex and side, but
  have no common interior points.

C
A
B
D
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Angles and Their Measures

• Example 4
• Use a protractor to draw two adjacent angles ?LMN
  and ?NMO so that ?LMN is acute and ?LMO is
  straight.

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Unit 1-Basics of Geometry

• 1.5 Segment and Angle Bisectors

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Segment and Angle Bisectors

• Midpoint
• The point on the segment that is the same
  distance from both endpoints.
• This point bisects the segment.
• Bisect
• To cut in half (two equal pieces).

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Segment and Angle Bisectors

• M is the midpoint of LN
• L M N
• LM ? MN

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Segment and Angle Bisectors

• Segment bisector
• A segment, ray, line, or plane that intersects a
  segment at its midpoint.

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Segment and Angle Bisectors

• Compass
• Geometric tool that is used to construct circles
  and arcs.
• Straightedge
• Ruler without marks.
• Construction
• Geometric drawing that uses a compass and
  straightedge.

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Segment and Angle Bisectors

• Construct a Segment Bisector and Midpoint
• Use the following steps to construct a bisector
  of AB and find the midpoint M of AB.
• Place the compass point at A. Use a compass
  setting greater than half of AB. Draw an arc.
• Keep the same compass setting. Place the compass
  point at B. Draw an arc. It should intersect
  the other arc in two places.
• Use a straightedge to draw a segment through the
  points of intersection. This segment bisects AB
  at M, the midpoint of AB.

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Segment and Angle Bisectors

• Midpoint Formula
• Given two points (x1, y1) and (x2, y2) the
  coordinates of the midpoint are
• x1 x2 , y1 y2
• 2 2

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Segment and Angle Bisectors

• Example 1
• Find the coordinates of the midpoint of the
  segment with endpoints at (12, -8) and (-3, 15).

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Segment and Angle Bisectors

• Example 2
• Find the coordinates of the midpoint of the
  segment with endpoints at (5, 8) and (7, -2).

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Segment and Angle Bisectors

• Example 3
• One endpoint is (17,-3) and the midpoint is
  (8,2).
• Find the coordinates of the other endpoint.

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Segment and Angle Bisectors

• Example 4
• One endpoint is (-5,8) and the midpoint is (6,3).
  Find the coordinates of the other endpoint.

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Segment and Angle Bisectors

• Angle bisector
• A ray that divides an angle into two adjacent
  angles that are congruent.

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Segment and Angle Bisectors

• Construct an Angle Bisector
• Place the compass point at C. Draw an arc that
  intersects both sides of the angle. Label the
  intersections A and B.
• Place the compass point at A. Draw another arc.
  Then place the compass point at B. Using the
  same compass setting, draw a third arc to
  intersect the second one.
• Label the intersection D. Use a straightedge to
  draw a ray from C through D. This is the angle
  bisector.

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Segment and Angle Bisectors

• Example 5
• JK bisects ?HJL. Given that m?HJL 42, what
  are the measures of ?HJK and ?KJL?

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Segment and Angle Bisectors

• Example 6
• A cellular phone tower bisects the angle formed
  by the two wires that support it. Find the
  measure of the angle formed by the two wires.

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Segment and Angle Bisectors

• Example 7
• MO bisects ?LMN. The measures of the two
  congruent angles are (3x 20) and (x 10) .
  Solve for x.

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Unit 1-Basics of Geometry

• 1.6 Angle Pair Relationships

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Angle Pair Relationships

• Vertical Angles
• Angles whose sides form opposite rays.

?1 and ?3 are vertical angles. ?2 and ?4 are
vertical angles.
1
4
2
3
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Angle Pair Relationships

• Linear Pair of Angles
• Angles that share a common vertex and a common
  side. Their non-common sides form a line.
• ?5 and ?6 are a linear pair of angles.

5
6
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Angle Pair Relationships

• Example 1
• Are ?1 and ?2 a linear pair?
• Are ?4 and ?5 a linear pair?
• Are ?5 and ?3 vertical angles?
• Are ?1 and ?3 vertical angles?

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Angle Pair Relationships

• Example 2

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Angle Pair Relationships

• Example 3
• Solve for x and y. Then find the angle measures.

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Angle Pair Relationships

• Complementary Angles
• Two angles that have a sum of 90º
• Each angle is a complement of the other.

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Angle Pair Relationships

• Supplementary Angles
• Two angles that have a sum of 180º
• Each angle is a supplement of the other.

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Angle Pair Relationships

• Example 4
• State whether the two angles are complementary,
  supplementary or neither.
• The angles formed by the hands of a clock at
  1100 and 100.

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Angle Pair Relationships

• Example 5
• Given that ?G is a supplement of ?H and m?G is
  82, find m?H.
• Given that ?U is a complement of ?V, and m?U is
  73, find m?V.

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Angle Pair Relationships

• Example 6
• ?T and ?S are supplementary.
• The measure of ?T is half the measure of ?S.
  Find m?S.

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Angle Pair Relationships

• Example 7
• ?D and ?E are complements and ?D and ?F are
  supplements. If m?E is four times m?D, find the
  measure of each of the three angles.

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Unit 1-Basics of Geometry

• 1.7 Introduction to Perimeter, Circumference,
  and Area

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Introduction to Perimeter, Circumference, and Area

• Square
• Side length s
• P 4s
• A s2

s
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Introduction to Perimeter, Circumference, and Area

• Rectangle
• Length l and width w
• P 2l 2w
• A lw

l
w
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Introduction to Perimeter, Circumference, and Area

• Triangle
• Side lengths a, b, and c,
• Base b, and height h
• P a b c
• A ½bh

a
c
h
b
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Introduction to Perimeter, Circumference, and Area

• Circle
• Radius r
• C 2p r
• A p r2
• Pi (p) is the ratio of the circles circumference
  to its diameter. p 3.14

r
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Introduction to Perimeter, Circumference, and Area

• Example 1
• Find the perimeter and area of a rectangle of
  length 4.5m and width 0.5m.

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Introduction to Perimeter, Circumference, and Area

• Example 2
• A road sign consists of a pole with a circular
  sign on top. The top of the circle is 10 feet
  high and the bottom of the circle is 8 feet high.
  
• Find the diameter, radius, circumference and area
  of the circle. Use p 3.14.

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Introduction to Perimeter, Circumference, and Area

• Example 3
• Find the area and perimeter of the triangle
  defined by H(-2, 2), J(3, -1), and K(-2, -4).

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Introduction to Perimeter, Circumference, and Area

• Example 4
• A maintenance worker needs to fertilize a 9-hole
  golf course. The entire course covers a
  rectangular area that is approximately 1800 feet
  by 2700 feet. Each bag of fertilizer covers
  20,000 square feet. How many bags will the
  worker need?

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Introduction to Perimeter, Circumference, and Area

• Example 5
• You are designing a mat for a picture. The
  picture is 8 inches wide and 10 inches tall. The
  mat is to be 2 inches wide. What is the area of
  the mat?

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Introduction to Perimeter, Circumference, and Area

• Example 6
• You are making a triangular window. The height
  of the window is 18 inches and the area should be
  297 square inches. What should the base of the
  window be?