Download Section 10.3 – Polygons, Perimeter, and Tessellations – pg 126

126
Section 10.3 – Polygons, Perimeter, and Tessellations
Objective #1:
Understanding & Classifying Different Types of Polygons.
A Polygon is closed two-dimensional geometric figure consisting of at least
three line segments for its sides. If all the sides are the same length and all
the angles have the same measure, then the figure is called a Regular
Polygon. The points where the two sides intersect are called vertices.
Triangle
3 sides
Quadrilateral
4 sides
Pentagon
5 sides
Hexagon
6 sides
Octagon
8 sides
Nonagon
9 sides
Decagon
10 sides
Dodecagon
12 sides
Heptagon
7 sides
Regular Polygons
Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
Nonagon
Decagon
Dodecagon
Heptagon
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Objective #2: Understanding properties of various quadrilaterals.
Parallelogram Rectangle
A
B E
F
C
D
Properties:
AB || CD
AC || BD
AB = CD
AC = BD
m∠A = m∠D
m∠B = m∠C
Objective #3:
G
H
Properties:
EF || GH
EG || FH
EF = GH
EG = FH
All the angles
measure 90˚.
The diagonals
EH and GF
are equal.
Square
I
Rhombus
M
N
J
K
L
Properties:
IK || JL
IJ || KL
All sides are
equal.
All the angles
measure 90˚.
The diagonals
IL and KJ
are equal.
O
Trapezoid
Q
R
P
S
Properties:
MN || OP
MO || NP
All sides are
equal.
m∠M = m∠P
m∠N = m∠O
T
Properties:
Only two
sides are
parallel:
ST || QR
Finding the perimeter of a figure.
The Perimeter is the length around the outside of a closed two dimensional figure. For a polygon, the perimeter is the sum of the length of
the sides of the polygon. We use the idea of perimeter when we calculate
how much fencing we need to enclose a garden or the amount of wood we
need to frame in a door.
Find the perimeter of the following:
Ex. 1
4 ft
Ex. 2
4 ft
5 ft
9
13
6 ft
6 ft
Solution:
To find the perimeter, simply
add up the lengths of the sides:
P = 6 + 4 + 4 + 5 + 6 = 25 ft.
€
11
13
m
m
12
13
m
Solution:
€
To find the perimeter, simply
add up the lengths of the sides:
9
12
11
32
P=
m +€ m +
m=
m.
€
13
€
13
€
13
€
13
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Perimeter of a Triangle
In general, if a, b, and c are the lengths of the
sides of a triangle, then the formula for the
perimeter of a triangle is P = a + b + c.
a
b
c
Find the perimeter of the following:
Ex. 3
Ex. 4
7.5 m
3.8 m
3.8 m
9 yd
9 yd
7.5 m
9 yd
9 yd
Solution:
To find the perimeter, simply
add up the lengths of the sides:
P = 2(7.5) + 2(3.8) = 22.6 m.
Solution:
To find the perimeter, simply
add up the lengths of the sides:
P = 4(9) = 36 yd.
Perimeter of a Rectangle
In general, if L is the length of a rectangle
and w is the width of the rectangle, then the
formula for the perimeter of a rectangle is
P = 2L + 2w.
Perimeter of a Square
In general, if s is the length of the side of
a square, the formula for the perimeter of
a square is P = 4s.
w
L
s
Find the following:
Ex. 5
A rectangular playground is 80 feet by 50 feet. If the fencing
costs $14 per yard, how much will it cost to enclose the playground?
Solution:
First, find the perimeter of the playground:
P = 2(80) + 2(50) = 160 + 100 = 260 feet
Next, convert the 260 feet into the yards:
260 ft 1yd
260
260 ft =
•
=
yd = 86.6666… yd
1
3 ft
3
Now, treat the $14 per yard as a conversion:
86.6666...yd $14
86.6666… yd =
•
≈ $1213.33
yd
1
€
€
€
€
€
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Objective #4:
Find the sum of the angles of a polygon.
Find the sum of the measures of the angles of the polygons below:
Ex. 6
Ex. 7
Solution:
We can split any quadrilateral into two triangles:
Solution:
We can split any pentagon
into three triangles:
The sum of the angles
The sum of the angles
in each triangle is 180˚, so
in each triangle is 180˚, so
the sum for quadrilateral is
the sum for pentagon is
2•180˚ = 360˚.
3•180˚ = 540˚.
In general, the number of triangles needed is two less than the number of
sides of the polygon. This leads to following formula:
Sum of the Angles of a Polygon
The sum of the measures of the angles of a polygon with n sides is
Sum = (n – 2)180˚
Solve the following:
Ex. 8a
Find the sum of the measures of the angles of a heptagon.
Ex. 8b
Find the measure of each angle of a regular heptagon.
Solution:
a)
Since a heptagon has 7 sides, then the sum of the angles of a
heptagon is (7 – 2)180˚ = (5)180˚ = 900˚
b)
All the angles will have the same measure in a regular
heptagon. Thus, the measure of each angle will be:
4
900˚ ÷ 7 = 128 ˚
7
Ex. 9
Find the measure of ∠B in
the diagram to the right.
Assume the€figure is a
regular polygon.
B
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Solution:
The sum of the measures of the angles of hexagon is (6 – 2)180˚ =
720˚. Since the hexagon is a regular polygon, each angle has a
measure of 720˚ ÷ 6 = 120˚.
∠B is the supplement of the angle on the bottom right of the hexagon.
Thus, m∠B = 180˚ – 120˚ = 60˚.
Objective #5: Understanding the angle requirements for tessellations.
A tessellation of a flat surface is the tiling of the surface using one or more
geometric shapes, called tiles, in a regular pattern leaving no overlaps and
no gaps. This is often used to tile a floor or wall with mosaics. Some
examples of tessellations using regular polygons are illustrated below:
Squares
Equilateral
Triangles
Regular Octagons
& Squares
Regular
Hexagons
Regular Hexagons
& Equilateral Triangles
Equilateral Triangles
& Squares
At each vertex in a tessellation, the same number and types of polygons
must be used and the sum of the measures of the angles must equal to
360˚. Otherwise, it will not be a tessellation.
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Show that each figure is a tessellation:
Ex. 10
Solution:
The sum of the measures of the angles of a hexagon is 4(180˚)
= 720˚. Thus, each angle is 720˚ ÷ 6 = 120˚. Since each vertex is
surrounded by three regular hexagons and the sum of the measures
of the angles at each vertex is 120˚ + 120˚ + 120˚ = 360˚, then the
figure is a tessellation.
Ex. 11
Solution:
The sum of the measures of the angles of an octagon is 6(180˚)
= 1080˚. Thus, each angle is 1080˚ ÷ 8 = 135˚. Each vertex is
surrounded by two regular octagons and one square. Also, the sum
of the measures of the angles at each vertex is
135˚ + 135˚ + 90˚ = 360˚, so the figure is a tessellation.
Ex. 12
Explain why a tessellation cannot be formed using regular
decagons only.
Solution:
The sum of the measure of the angles of decagon is 8(180˚) = 1440˚.
Each angle of a regular decagon is 1440˚ ÷ 10 = 144˚. Two regular
decagons fill in 2•144˚ = 288˚, leaving a gap of 360˚ – 288˚ = 72˚.
There can be no gaps with tessellations so a tessellation cannot be
formed with using only regular decagons.