CHAPTER2: CHAPTER2: Computer Architecture Foundation EKT303/4.
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Transcript of chapter2-polygon.pdf
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Mohd Khusaini MajidMrsm Kota Kinabalu
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2.1 CONCEPT OF REGULAR
POLYGONS
An equilateral triangle has equal sides and equal interior angles. Thus, AB = BC = CA and
A = B = C = 60
A
A
A
C B
D C
B
E
D C
B
A square has equal sides and equal interior angles. Thus, AB = BC = CD = DA and
A = B = C = D = 90
Is all sides of pentagon ABCDEhas the same length and the angles are of the same size?
RECALL
The sum of the interior angles of a triangle is 180.
The sum of the interior angles of a square is 360.
A regular polygon is a polygon in which
a) all sides are of equal length and
b) all interior angles are of equal size.
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Example 1Example 1Example 1Example 1Determine if each of the polygons below is a regular polygon. Give your reason if it is not a regular polygon.
a) A
D C
BSolution:
ABCD is not a regular polygon because A B.
b) BAH
G
F
C
D
E
Solution:
ABCDEFGH is a regular polygon.
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Test YourselfTest YourselfTest YourselfTest YourselfDetermine if the following are regular polygon. Give your reason if it is not a regular polygon.
1. L
M N
4.3.
2. SP
RQ
U
T
S
R VW
VU
T
3 cm
3 cm 3 cm
3 cm
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Copy the following polygons. Draw all the axes of symmetry of each polygon if there are any. State the number of axes of each polygon.
Exercise 2.1AExercise 2.1AExercise 2.1AExercise 2.1A
1.
4.3.
2.
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Find the size of interior and exterior angles
2.2 EXTERIOR AND INTERIOR
ANGLES OF POLYGONS
exterior angle
interior angle
In a polygon, the interior and exterior angles lie on a straight line.
Interior angle + Exterior angle = 180
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Example 4Example 4Example 4Example 4Find the values of x and y in the following polygons.a)
x
105
2y y
Solution:
x + 105 = 180
x = 180 105
= 75
2y + y = 180
3y = 180
y = 180
= 603
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b)
1002x
110
y
Solution:
2x + 100 = 180
2x = 180 100
= 80
x = 80
= 40
y + 110 = 180
y = 180 - 110
= 70
2
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Find the values of the unknown angles in each polygons below.
Exercise 2.2AExercise 2.2AExercise 2.2AExercise 2.2A
1.b
48
a132
2.110
75c
d
f
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Determine the sum of the interior angles of a polygon
RECALL
The sum of the interior angles of a triangle is 180.
The sum of the interior angles of a square is 360.
What is the sum of the interior angles of a pentagon, hexagon and other polygons?
The sum of the interior angles of a polygon with n sides is (n 2) x 180
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Example 5Example 5Example 5Example 5Find the value of x in each of the polygons below.a)
95
x
120110
Solution:
The sum of the interior angles of a pentagon = (5 2) x 180= 3 x 180
= 540
x + 90 + 120 + 95 + 110 = 540
x + 415 = 540
x = 540 415 = 125
Use (n 2) x 180
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b)140
x
x
85136125
Solution:
The sum of the interior angles of a hexagon = (6 2) x 180= 4 x 180
= 720
x + x + 140 + 125 + 136 + 85 = 720
2x + 486 = 720
2x = 720 486
x = 234 = 1172
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Example 6Example 6Example 6Example 6
Find the number of sides of a polygon if the sum of its interior angles is(a) 1440 (b) 1080
(a) Let n be the number of sides of a polygon.
(n 2) x 180 = 1440n 2 = 1440
= 8
n = 10
(b) Let n be the number of sides of the polygon
(n 2) x 180 = 1080n 2 = 1080
= 6
n = 8
Solution:
180 180
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Exercise 2.2BExercise 2.2BExercise 2.2BExercise 2.2B
1. Find the sum of the interior angles of each of the following polygons.
a) Pentagonb) Heptagonc) Decagon
2. Find the number of sides of a polygon if the sum of its interior angles is
a) 720b) 900c) 1260
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3. Find the value of x in each of the polygons below.a)
130x
140
135144
160
b)60
x
x
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4. The diagram below shows a hexagon. Find the value of x + y.
70
y
yx
x
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Determine the sum of the exterior angles of a polygon
The sum of the exterior angles of a polygon is 360.
B
A
D
C
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Example 7Example 7Example 7Example 7Find the values of the unknown angles in each of the polygons below.a)
40
y z
x 75
Solution:
x = 180 75
= 105
y = 360 (40 + 90 + 105)= 360 235
= 125
z = 180 125
= 55
Supplementary angles
Sum of the exterior angles of a polygon is 360
Supplementary angles
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b)
75
y
x3x
6560
E D
C
BA
Solution:
x + 3x = 180
4x = 180
x = 45
Extend the side EA.
Exterior angle of A = 180 75
= 105
y = 360 (60 + 45 + 105 + 65)= 360 275
= 85 Sum of the exterior angles of a polygon is 360
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Exercise 2.2CExercise 2.2CExercise 2.2CExercise 2.2C
1. Calculate the unknown angles in the following polygons.
a) b)112
45
6080
75
150
x
x
x
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c) d)
11074140
150100
68
75 y
x
z
w
zr
s
r
p
q
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Find the interior angles, exterior angles and number of sides of a regular polygon
A regular polygon has equal interior angles, equal exterior angles and sides of equal length.
The sum of the interior angles of a polygon with n sides is (n 2) x 180.
Thus, each interior angle of a regular polygon is (n 2) x 180
n
The sum of the exterior angles of a polygon is 360.
Thus, each exterior angle of a polygon is 360
n
Notes
If exterior angle = 360 , then
interior angle = 180 - 360 .n
n
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Example 8Example 8Example 8Example 8
Find the size of the interior angle and the exterior angle of a regular heptagon.Solution:
A regular heptagon has 7 sides.
Sum of the interior angles = (7 2) x 180= 900
Each interior angle = 900
= 128 4
7
7
Each exterior angle = 360
= 51 37
7
ANOTHER WAY: Exterior angle = 180 Interior angle
= 180 128 4 = 51 37 7
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Example 9Example 9Example 9Example 9Find the number of sides of a regular polygon given that(a) the exterior angle is 72 (b) the interior angle is 140
Solution:
(a) Let n be the number of sides of the polygon.360 = 72
Thus, n = 360
= 5
n
72
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(b) Let n be the number of sides of the polygon.(n 2) x 180 = 140
180n 360 = 140n180n - 140n = 360
40n = 360Thus, n = 360
= 9
n
Another Way: Interior angle = 140
Exterior angle = 180 - 140
= 40
Hence, 360 = 40
n = 9n
40
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Exercise 2.2DExercise 2.2DExercise 2.2DExercise 2.2D1. Find the size of the interior and exterior angles of the
following regular polygons.a) Pentagonb) Octagonc) Hexagond) Decagon
2. Find the number of sides of a regular polygon, given that its
a) interior angle is 135b) interior angle is 108c) exterior angle is 36d) exterior angle is 120
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Solve problems involving angles and sides of polygons
Example 10Example 10Example 10Example 10
Amin is given a square tile and two regular hexagonal tiles. All the tiles have sides of equal length. Determine if he can form a tessellation with these tiles. If Amin must use the square tile, find two other tiles which can tessellate with the square tile.
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Solution:
Understand the problemGiven : One square and two hexagons with sides of
the same lengthFind : Sum of one interior angle of a square and one
interior angle of each hexagon
Devising a strategyFind the interior angles of the three polygons.Add to see if the sum of the three interior angles mentioned above 360.
Stage 1
Stage 2
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Carrying out the strategyInterior angle of a square is 90.Interior angle of a hexagon is 180 - 360 = 120
Sum of interior angles of the square and two hexagons is 90 + (2 x 120) = 330. Thus, the three tiles do not tessellate.If Amin has to use the square tile and needs to find two tiles which can tessellate with it, each interior angle of the other two tiles is 360 - 90 = 135.
6
2
Stage 3
The sum of the interior angles of the square tile and the two other tiles must be 360. Thus, 90 + (2 x interior angle) = 360
Interior angle = 360 - 902
-
(n 2) x 180 = 135
180n 360 = 135n45n = 360
n = 360 = 8
Thus, the other two tiles should be in the shape of an octagon.
Checking the answerUse the strategy of working backwards.If two octagonal tiles are used, each interior angle is 135.Sum of the two interior angles of the two tiles is 2 x 135 = 270.To tessellate, the interior angle of the third polygon is 360 - 270 = 90. Thus, a square tile is needed to tessellate with two octagonal tiles.
n
45
Stage 4
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Exercise 2.2EExercise 2.2EExercise 2.2EExercise 2.2E1. In the diagram, ABCD is part of a regular decagon.
FBCG is part of a regular polygon. Calculatea) the number of sidesb) the sum of the interior anglesof the regular polygon FBCG.
GFD
CB
A
4
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SUMMARY POLYGONS IIPOLYGONS IIPOLYGONS IIPOLYGONS II
Regular polygon
A polygon in which all the sides are of equal length and all the interior angles are of equal size
Irregular polygon
A polygon in which not all the sides are of equal length or not all the interior angles are of equal size
Equilateral triangle
Square Regular pentagon
Regular hexagon
Scalene triangle Rectangle Parallelogram
Exterior angle and interior angle
Interior angle + Exterior angle = 180
The sum of the exterior angles of any polygon is 360.
The sum of the interior angles of a polygon with n sides is (n 2) x 180.
interior angle
exterior angle
The interior angle of a regular polygon with n sides is (n - 2) x 180 .
The exterior angle of a regular polygon with n sides is 360 .
n
n
Axis of symmetry
The number of axes of symmetry of a regular polygon is equal to its number of sides.