Welcome to MM150 Unit 6 Seminar. Line AB AB Ray AB AB Line segment AB AB.
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Transcript of Welcome to MM150 Unit 6 Seminar. Line AB AB Ray AB AB Line segment AB AB.
Welcome to MM150 Unit 6
Seminar
• Line AB AB
• Ray AB AB
• Line segment AB AB
Plane1. Any three points that do not lie on the
same line determine a plane. (Since 2 points determine a line, a line and a point not on the line determine a unique plane).
2. A line in a plane divides the plane into 3 parts: the line and 2 half-planes.
3. The intersection of 2 planes is a line.
3 Definitions• Parallel planes – 2 planes that do not
intersect
• Parallel lines – 2 lines IN THE SAME PLANE that do not intersect
• Skew lines – 2 lines NOT IN THE SAME PLANE that do not intersect.
Angle
A
D
F
Vertex
Side
Side
Angle MeasuresAcute Angle 0 degrees < acute < 90 degrees
Right Angle 90 degrees
Obtuse Angle 90 degrees < obtuse < 180 degrees
Straight Angle 180 degrees
More Angle Definitions
B D H
L M
2 angles in the same plane are adjacent angles if they have a common vertex and a common side, but no common interior points. Example: [ang]BDL and [ang]LDM Non-Example: [ang]LDH and [ang]LDM
2 angles are complementary angles if the sum of their measures is 90 degrees.Example: [ang]BDL and [ang]LDM
2 angles are supplementary angles if the sum of their measures is 180 degrees.Example: [ang]BDL and [ang]LDH
If the measure of [ang]LDM is 33 degrees, find the measures of
the other 2 angles.
B D H
L M
Given information:[ang]BDH is a straight angle[ang]BDM is a right angle
If the measure of [ang]LDM is 33 degrees, find the measures of
the other 2 angles.
B D H
L M
Given information:[ang]BDH is a straight angle[ang]BDM is a right angle
[ang]BDM=90
[ang]BDL=90-33=57 deg[ang]MDH=90 deg
If [ang]ABC and [ang]CBD are complementary and [ang]ABC is 10 degrees less than [ang]CBD, find the measure of both angles.
B A
CD
[ang]ABC + [ang]CBD = 90Let x = [ang]CBDThen x – 10 = [ang]ABC
X + (x – 10) = 902x – 10 = 902x = 100X = 50 [ang]CBD = 50 degreesX – 10 = 40 [ang]ABC = 40 degrees
Polygons# of Sides Name
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
9 Nonagon
10 Decagon
12 Dodecagon
20 Icosagon
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• The sum of the measures of the interior angles of a n-sided polygon is
• (n - 2)*180 degrees
What is the sum of the measures of the interior angles of a nonagon?
n = 9 (9-2) * 180 = 7 * 180 = 1260 degrees
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Sum of Interior Angles
2 * 180 = 360 degrees
3 * 180 = 540 degrees
4 * 180 = 720 degrees
4 - 2 = 2
5 - 2 = 3
6 - 2 = 4
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EVERYONE: How many sides does a polygon have if thesum of the interior angles is 900 degrees?
Formula:
(n - 2)*180 degrees, where n is number of sides of polygon
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EVERYONE: How many sides does a polygon have if thesum of the interior angles is 900 degrees?
• (n - 2) * 180 = 900
• Divide both sides by 180• n - 2 = 5
• Add 2 to both sides• n = 7 The polygon has 7 sides.
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Similar Figures
A
B
C X
Y
Z
80[deg]
80[deg]
50[deg] 50[deg]50[deg]50[deg]
[ang]A has the same measure as [ang]X[ang]B has the same measure as [ang]Y[ang]C has the same measure as [ang]Z
XY = 4 = 2AB 2
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1 2
4 4
YZ = 4 = 2BC 2
XZ = 2 = 2AC 1
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Page 238 # 73
• Steve is buying a farm and needs to determine the height of a silo. Steve, who is 6 feet tall, notices that when his shadow is 9 feet long, the shadow of the silo is 105 feet long. How tall is the silo?
6 ft
9 ft
105 feet
?
9 = 6105 ?
9 * ? = 105 * 6
9 * ? = 630
? = 70 feet
The silo is 70 feet tall.
Units in measurementLet’s consider a rectangle with length 5
inches and width 3 inches:
Perimeter = 2l + 2w = 10 in + 6 in = 16 inches
Area = l * w = 5 in * 3 in = 15 in*in = 15 in^2 (or 15 sq. in.)
Rectangular box with height 2 inches:
Volume = l * w * h = 5 in * 3 in * 2 in = 30 in*in*in = 30 in^3 (30 cubic inches)
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Area of a Trapezoid
2 m
3 m
4 m
A = (1/2)h(b1 + b2)
A = (1/2)(2)(3 + 4)
A = (1/2)(2)(7)
A = 1(7)
A = 7 square meters
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Circleradius is in greendiameter is in blue
2r = d Twice the radius is the diameter
CircumferenceC = 2∏r or 2r∏
Since 2r = dC = ∏d
AreaA = ∏r2
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Prisms
Pyramids
Volume• In 3 dimensions, so general rule is that
volume is base (area) times height (length)• For prisms V=Bh• For pyramids V=(1/3)Bh• Similarly with cylinders and cones• Page 255• Spheres• V = (4/3) pi * r^3• SA = 4 pi * r^2
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Examples
Page 263 #8Rectangular prism (box)
V = Bh
V = (6 sq yd)*(6 yard)
V = 36 cubic yards
Page 263 #14cone
V = (1/3)Bh
V = (1/3)(78.5 sq ft)(24 ft)
V = 628 cubic feet
Page 263 #16sphere
V = (4/3)pi*r^3
V = (4/3)(3.14)(7 mi)^3
V = 1436 cubic miles (approx.)
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Surface Area
• Remember surface area is the sum of the areas of the surfaces of a three-dimensional figure.
• Take your time and calculate the area of each side.
• Look for sides that have the same area to lessen the number of calculations you have to perform.
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Examples of surface area
Page 263 #8
Area of the 2 Bases3 yd * 2 yd = 6 sq yd
Area of 2 sides2 yd * 6 yd = 12 sq yd
Area of other 2 sides3 yd * 6 yd = 18 sq yd
Surface area6 + 6 + 12 + 12 + 18 + 18
= 72 sq yd
Page 263 #14
Surface area of a coneSA = [pi]r2 + [pi]r*sqrt[r2 + h2]
SA = 3.14 * (5)2 + 3.14 * 5 * sqrt[52 + 242]
SA = 3.14 * 25 + 3.14 * 5 * sqrt[25 + 576]
SA = 78.5 + 15.7 sqrt[601]
SA = 78.5 + 24.5
SA = 103 sq ft
