introduction to geometry
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INTRODUCTION TO GEOMETRY. MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur. Geometry. The word geometry comes from Greek words meaning “to measure the Earth” Basically, Geometry is the study of shapes and is one of the oldest branches of mathematics. - PowerPoint PPT PresentationTRANSCRIPT
INTRODUCTION TO GEOMETRY
MSJC ~ San Jacinto CampusMath Center Workshop Series
Janice Levasseur
Geometry
• The word geometry comes from Greek words meaning “to measure the Earth”
• Basically, Geometry is the study of shapes and is one of the oldest branches of mathematics
The Greeks and Euclid
• Our modern understanding of geometry began with the Greeks over 2000 years ago.
• The Greeks felt the need to go beyond merely knowing certain facts to being able to prove why they were true.
• Around 350 B.C., Euclid of Alexandria wrote The Elements, in which he recorded systematically all that was known about Geometry at that time.
Basic Terms & Definitions
• A ray starts at a point (called the endpoint) and extends indefinitely in one direction.
• A line segment is part of a line and has two endpoints.
A B AB
BA AB
• An angle is formed by two rays with the same endpoint.
• An angle is measured in degrees. The angle formed by a circle has a measure of 360 degrees.
vertex
side
side
• A right angle has a measure of 90 degrees.
• A straight angle has a measure of 180 degrees.
• A simple closed curve is a curve that we can trace without going over any point more than once while beginning and ending at the same point.
• A polygon is a simple closed curve composed of at least three line segments, called sides. The point at which two sides meet is called a vertex.
• A regular polygon is a polygon with sides of equal length.
Polygons
# of sides name of Polygon3 triangle
4 quadrilateral5 pentagon6 hexagon7 heptagon8 octagon9 nonagon10 decagon
Quadrilaterals• Recall: a quadrilateral is a 4-sided polygon. We can
further classify quadrilaterals: A trapezoid is a quadrilateral with at least one pair of
parallel sides. A parallelogram is a quadrilateral in which both pairs of
opposite sides are parallel. A kite is a quadrilateral in which two pairs of adjacent
sides are congruent. A rhombus is a quadrilateral in which all sides are
congruent. A rectangle is a quadrilateral in which all angles are
congruent (90 degrees) A square is a quadrilateral in which all four sides are
congruent and all four angles are congruent.
From General to Specific
Quadrilateral
trapezoidkite
parallelogram
rhombus
rectangle
square
Mo
re s
pe
cific
Perimeter and Area
• The perimeter of a plane geometric figure is a measure of the distance around the figure.
• The area of a plane geometric figure is the amount of surface in a region.
perimeter
area
Triangleh
b
a c
Perimeter = a + b + c
Area = bh21
The height of a triangle is measured perpendicular to the base.
Rectangle and Square
w
l
s
Perimeter = 2w + 2l Perimeter = 4s
Area = lw Area = s2
Parallelogram
b
a h
Perimeter = 2a + 2b
Area = hb Area of a parallelogram = area of rectangle with width = h and length = b
Trapezoid
c d
a
bPerimeter = a + b + c + d
Area =
b
a
Parallelogram with base (a + b) and height = h with area = h(a + b) But the trapezoid is half the parallelgram
h(a + b)21
h
Ex: Name the polygon
3
21
45
6
hexagon
1
2
34
5 pentagon
Ex: What is the perimeter of a triangle with sides of lengths 1.5
cm, 3.4 cm, and 2.7 cm?
1.5 2.7
3.4
Perimeter = a + b + c
= 1.5 + 2.7 + 3.4
= 7.6
Ex: The perimeter of a regular pentagon is 35 inches. What is the
length of each side?
Perimeter = 5s
35 = 5s
s = 7 inches
s
Recall: a regular polygon is one with congruent sides.
Ex: A parallelogram has a based of length 3.4 cm. The height
measures 5.2 cm. What is the area of the parallelogram?
3.4
5.2
Area = (base)(height)
Area = (3.4)(5.2)
= 17.86 cm2
Ex: The width of a rectangle is 12 ft. If the area is 312 ft2, what is the length of the rectangle?
12 312
Area = (Length)(width)
L = 26 ft
Let L = Length
L312 = (L)(12)
Check: Area = (Length)(width) = (12)(26)
= 312
Circle
• A circle is a plane figure in which all points are equidistance from the center.
• The radius, r, is a line segment from the center of the circle to any point on the circle.
• The diameter, d, is the line segment across the circle through the center. d = 2r
• The circumference, C, of a circle is the distance around the circle. C = 2r
• The area of a circle is A = r2.
r
d
Find the Circumference
• The circumference, C, of a circle is the distance around the circle. C = 2r
• C = 2r• C = 2(1.5)• C = 3cm
1.5 cm
Find the Area of the Circle• The area of a circle is A = r2
• d=2r• 8 = 2r• 4 = r
• A = r2
• A = 2
• A = 16sq. in.
8 in
Composite Geometric Figures
• Composite Geometric Figures are made from two or more geometric figures.
• Ex:
+
• Ex: Composite Figure
-
Ex: Find the perimeter of the following composite figure
+=
8
15
Rectangle with width = 8 and length = 15
Half a circle with diameter = 8 radius = 4
Perimeter of composite figure = 38 + 4.
Perimeter of partial rectangle = 15 + 8 + 15 = 38
Circumference of half a circle = (1/2)(24) = 4.
Ex: Find the perimeter of the following composite figure
28
60
42
12
? = a
? = b
60
a 42
60 = a + 42 a = 18
28
b
12
28 = b + 12 b = 16
Perimeter = 28 + 60 + 12 + 42 + b + a = 28 + 60 + 12 + 42 + 16 + 18 = 176
Ex: Find the area of the figure
3
3
8
8
Area of rectangle = (8)(3) = 24
3
8
Area of triangle = ½ (8)(3) = 12
Area of figure = area of the triangle + area of the square = 12 + 24 = 36.
3
Ex: Find the area of the figure4
3.5
4
3.5
Area of rectangle = (4)(3.5) = 14
4
Diameter = 4 radius = 2
Area of circle = 22 = 4 Area of half the circle = ½ (4) = 2
The area of the figure = area of rectangle – cut out area
= 14 – 2 square units.
Ex: A walkway 2 m wide surrounds a rectangular plot of grass.
The plot is 30 m long and 20 m wide. What is the area of the walkway?
20
302
What are the dimensions of the big rectangle (grass and walkway)?
Width = 2 + 20 + 2 = 24
Length = 2 + 30 + 2 = 34
The small rectangle has area = (20)(30) = 600 m2.
What are the dimensions of the small rectangle (grass)?
Therefore, the big rectangle has area = (24)(34) = 816 m2.
The area of the walkway is the difference between the big and small rectangles:
20 by 30
Area = 816 – 600 = 216 m2.
2
Find the area of the shaded region10
10
10 r = 5
Area of each circle = 52 = 25
¼ of the circle cuts into the square.
But we have four ¼
4(¼)(25) cuts into the area of the square.
Area of square = 102 = 100
Therefore, the area of the shaded region = area of square – area cut out by circles = 100 – 25 square units
r = 5