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Transcript of GEOMETRY CHAPTER 3. Geometry & Measurement 3.1 Measuring Distance, Area and Volume 3.2 Applications...
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GEOMETRY
CHAPTER 3
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Geometry & Measurement
3.1 Measuring Distance, Area and Volume
3.2 Applications and Problem Solving
3.3 Lines, Angles and Triangles
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3.1 Rounding Measurements
To round: 1. Underline the place 2. If number to the right of the under-lined place is 5 or more, add one 3. Otherwise, do not change4. Change all digits to the right of underlined number to zeros
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3.1 Rounding Example
1. 38.67
2. First number to the right of 8 is “6”, so add one to 8
4. Change all digits to the right to 0’s. The answer is, 39.00 or 39
Example: Round 38.67 centimeters to the nearest centimeter
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3.1 Calculating Distances
Linear Measure - a distance which could be around a polygon (perimeter) or around a circle (circumference)
Perimeter - sum of the lengths of the sides
C d r= =π (d Re member )2Circumference - distance around circle
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Measure can be in U.S. system (yd, ft, etc.) or metric (cm,m, etc)
3.1 Metric Measures
King
Milk
Henry Died
Drinking
Monday
Chocolate
km hm dam m
dm cm mm
kilometer hectometer dekameter meter
decimeter centimeter millimeter
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3.1 Metric Measures
1 cm = 0.01 m
1 dm = 0.1 m
1 mm = 0.001 m
1 hm = 100 m
1 km = 1000 m
1 dam = 10 m
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3.1 Linear Distance
2. What is the distance around the polygon, in meters?
75 cm78 cm
95 cm 80 cm 78 + 95 +
80 + 75 = 328 cm
km
A. 328 m
hm dam m dm cm mm
B. 32.8 m
C. 3.28 m D. 0.328m
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3.1 Calculating Areas
Rectangle
Parallelogram
Square
Triangle
Trapezoid
Circleb1
b2
r
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3.1 Area - Square Units
4. What is the area of a circular region whose diameter is 6 cm?
If d = 6,
then r = 3A = r 2π
Formula:
=π (3)2
€
D. 9π sq. cm
€
B. 6π sq. cm
€
A. 36π sq. cm
€
C. 12π sq. cm
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Surface area of a rectangular solid
3.1 Examples of Area
LW
H
There are 6 faces of the solid
A=2LH
Front/backSides (Left/Right)
+2WH
Top/Bottom
+2LW Square units
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3.1 Examples of Area
6. What is the surface area of a rectangular solid that is 12 in. long, 5 in. wide and 6 in. high?
D. 360 sq. in.
L=12W=5
H=6
A=2(12)(6)+2(5)(6)+2(12)(5)A=2LH +2WH +2LW
A. 360 cubic in. B. 324 sq. in.
C. 324 cubic in.
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3.1 Volume - Cubic Units
Rectangular Solid
Cylinder
h
h
h Cone
Sphere
V=lwh
V r h=π 2
hrV 2 3
1π=
V r=43
3π
r
rr
w l
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3.1 Example of Volume
8. What is the volume of a sphere with a 12 inch diameter?
Formula: V r=43
3π If d = 12,then r = 6
V =43
6 3π ( ) Since (6)(6)(6)= 216, the only reasonable ans. is C
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3.1 Identifying the Unit
9. Which of the following would not be used to measure the amount of water needed to fill a swimming pool?
A. Cubic feet
Think of “volume” as capacity or filling up the inside of a 3D figure.
linear
B. Liters
C. Gallons D. Meters
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3.2 Application Example
1. What will be the cost of tiling a room measuring 12 ft. by 15 ft. if square tiles cost $2 each & measure 12 in.?
Since 12 inches = 1 ft, one tile is 1 ft on each side or 1 sq. ft.Area room: A = bh; (12)(15) = 180 sq ft And (180)($2) = $360 cost
A. $180 B. $4320 C. $360 D. $3600
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3.2 Pythagorean Theorem
For any RIGHT TRIANGLE c
c a b2 2 2= +
a
b
Side opposite the right angle is the hypotenuse “c”
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3.2 Pythagorean Theorem
c a b2 2 2= +
3. A TV antenna 12 ft. high is to be anchored by 3 wires each attached to the top of antenna and to pts on the roof 5 ft. from base of the antenna. If wire costs $.75 per ft, what will be the cost?
12
€
c2 = (12)2 +(5)2 = 144 + 25 = 169
Cost is .75 x 39 =$29.25
5
c
€
c = 13 and 3 wires x 13 ft = 39 ft
A. $27.00 B. $29.25 C. $9.75 D. $38.25
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3.2 Infer & Select Formulas
7. The figure shows a regular hexagon Select the formula for total area
b
Total area is the area of the 6 identical triangles.
A. 3h+b
If area of 1 triangle = 1/2xbh,then 6 x 1/2 x bh = 3 bh
B. 6(h+b) C. 6hb D. 3hb
h
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3.3 Lines; Angles; Triangles
straight angle 180right angle 90obtuse > 90, < 180acute angle < 90comp. sum to 90supp. sum to 180vertical angles-equal
ANGLES TRIANGLES
Right triangle Acute triangle Obtuse triangle Scalene triangle Isosceles Equilateral
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3.3 Properties Example
2. What type of triangle is ABC?
55°
70°
A. Isosceles
Since sum of angles of triangle = 180,and 55 + 70 = 125,then angle C = 180 - 125 = 55.If 2 angles = , then isosceles. C
B. RightC. Equilateral D. Scalene
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3.3 Angle Measures
BB
BB
SS
SS
1. B S Theorem All B’s are = , All S’s are = B + S = 180
2. Perpendicular lines intersect to form right angles.
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3.3 Angle Measures
7
L11Terminology
The parallel lines are cut by transversal T
L2
45
8
32
6
Corresponding angles are =1 and 5, 3 and 7, 2 and 6, 4 and 8
Vertical angles are =1 and 4, 3 and 2, 6 and 7, 5 and 8
T
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3.3 Angle Measures
7
L11Terminology
The parallel lines are cut by transversal T
L2
45
8
32
6
TAlternate interior angles are =4 and 5, 3 and 6
Alternate exterior angles are =1 and 8, 2 and 7
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3.3 Angle Measures
3. If 2 angles of a triangle are = , then sides opposite are =
4. If 2 sides of a triangle are =, then angles opposite are =
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3.3 Examples
7575
75
75
10545
45
4545
135
6. Which statement is true for the figure shown at the right given that L1 and L2 are parallel?
After using the BS theorem, angle T does = 75 and angle S=105
€
A.Sincem∠T = 75°,m∠S = 60°
TSV
R
€
B.Sincem∠T = 75°,m∠S = 105°
€
C.m∠V = m∠R
€
D.None
L1
L2
135105
60
60
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3.3 Similar Triangles
Two triangles are similar if all angles are = and sides proportional
10.Which statements are true?i. m A = m E ii. AC = 6 iii. CE/CA = CB/CD
A. i only B. ii only C. i and ii only D. i, ii, iii
4040
7.5A
x
E5
CD
B4
Since m D=m B and DCE and ACB areVertical angles m A=m E
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3.3 Similar Triangles
Two triangles are similar if all angles are = and sides proportional
10.Which statements are true?i. m A = m E ii. AC = 6 iii. CE/CA = CB/CD
A. i only B. ii only C. i and ii only D. i, ii, iii
4040
7.5A
x
E5
CD
B4
The triangle are similar, thus ratios of corresponding sides are =. x/4 = 7.5/5 thus x= 4(7.5)/5 = 6
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3.3 Similar Triangles
Two triangles are similar if all angles are = and sides proportional
10.Which statements are true?i. m A = m E ii. AC = 6 iii. CE/CA = CB/CD
A. i only
4040
7.5A
x
E5
CD
B4
The triangle are similar, thus ratios of corresponding sides are =. CE/CA = CD/CB thus iii is false!
B. ii only C. i and ii only D. i, ii, iii
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REMEMBER
MATH IS FUN
AND …
YOU CAN DO IT