introduction to geometry msjc ~ san jacinto campus math center workshop series janice levasseur
TRANSCRIPT
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.
The Elements
• Knowing that you can’t define everything and that you can’t prove everything, Euclid began by stating three undefined terms:
Point (Straight) Line
Plane (Surface)
Actually, Euclid did attempt to define these basic terms . . .
is that which has no part
is a line that lies evenly with the points on itself
is a plane that lies evenly with the straight lines on itself
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 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
Mor
e sp
ecifi
c
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.
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
b
Perimeter = 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: 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 = LengthL
312 = (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: 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
10r = 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