Semester 2: Unit 7 TAXICAB GEOMETRY

GT/Honors Geometry
(Subject to change)
4/29 to June 5
Date
Wednesday
4/29
Thursday
4/30
Friday
5/1
Monday
5/4
Tuesday. 5/5
Wednesday. 5/6
Thursday
5/7
Friday
5/8
Topic
Taxicab geometry
TEST: CIRCLE THEOREM
Euclidean and non-Euclidean geometry
Compare and contrast.
Spherical Geometry (Page 154-155)
STAAR TEST – ALG 1
STAAR TEST – BIOLOGY
Spherical geometry Part 2
Major Quiz: 12-5 Equations of circles and
Taxicab Geometry.
Comparing Planar and Spherical
Geometry
Homework
WS -Taxicab Geometry
Worksheet Euclidean and Non Euclidean Geo
Worksheet Euclidean and Non Euclidean Geo
Page 155: #1-9
WS – Spherical geometry Part 2
WS - Comparing Planar and Spherical Geometry
Monday
5/11
Tuesday
5/12
Wednesday
5/13
Thursday
5/14
Friday
5/15
Coordinates in Space
Monday. 5/18
Tuesday. 5/19
Wednesday. 5/20
Thursday 5/21
Friday 5/22
SAT/ACT TOPICS
SAT/ACT TOPICS
SAT/ACT TOPICS
SAT/ACT TOPICS
SAT/ACT TOPICS
TBA
TBA
TBA
TBA
TBA
Monday 5/25
Tuesday
Wednesday
Thursday
Friday
Memorial Day - Holiday
TBA
TBA
TBA
TBA
TBA
Monday, June 01
Tuesday, June 02
Wednesday, June 03
Thursday, June 04
FINALS
FINALS
Coordinates in Space
Coordinates in space ws-1
Coordinates in space ws-2
Quiz: Coordinates in space, spherical
Geometry.
Study for test
Review for test
Study for test
Test: Euclidean and non-Euclidean
Geometry, Coordinates in Space, and
Equations of circles
Review
Review
Review
Review
FINALS
Early Dismissal
6th PERIOD EXAM
2nd and 4th PERIOD Exam
3rd and 5th PERIOD EXAM
1st and 7th PERIOD EXAM
Note
Taxicab Geometry
Taxicab geometry, introduced by Hermann Minkowski in the 19th century, is a form of geometry in which the
usual definition of distance in Euclidean geometry is replaced by a new metric in which the distance between two
points is the sum of the absolute differences of their coordinates.
In Plane (OR Euclidean) Geometry, the distance between points P(x1, y1) and Q(x2, y2) is given by
√
In Taxicab Geometry, the distance between points P(x1, y1) and Q(x2, y2) is given by
|
| |
|
A section of downtown Houston is shown on the right.
A cab is at the intersection of Elgin Street and Fannin Street.
If the cab driver wants to reach the intersection of McGowen Street
and Crawford Street to pick up a customer, he cannot drive in a
straight line. Although he can take different routes, the shortest
distance is given by the formula given above.
In taxicab geometry all three pictured lines (red, yellow, and blue) have the same length (12 units) for the same
route. In Euclidean geometry, the green line has length √
≈ 8.48, and is the unique shortest path
Application
A dispatcher for a City Police Department receives a report of an accident at X = (-6,4). There are two
police cars located in the area. Officer Hall is at (2,1) and Officer Lin is at (-1,-6). Which car should be
sent to the accident site?
Ws
Taxicab Geometry Questions: Name______________________Date__________Period_______
2. Draw the “taxi” circle with center at (1, -2) and
y
radius 5.
1. Find the taxicab distance between (-1, 2) and
y
(3, 5).
x
3. Find the midpoint of the segment with
endpoints at (-6, -5) and (7, 4).
y
x
4. Find the perpendicular bisector of the segment
with endpoint at (-3, -5) and (5, 1). y
x
x
Apartment hunting – Jane and George are looking for an apartment in a city where the streets all follow
the grid lines. Jane works as a waitress at a restaurant at J(-3, -1). George works as a technician at the
local television station at G(3,3). They walk wherever they go.
5. They have decided their apartment should be
located so that the distance Jane has to walk to
work plus the distance George has to walk to
work is as small as possible. Where should they
look for an apartment?
y
6. In a moment of chivalry George decides that
the sum of the distances should still be a
minimum, but Jane should not have to walk any
farther than he does. Now where could they look
y
for an apartment?
x
x
7. Jane agrees that the sum of the distances
should be a minimum, but she is adamant that
they both have exactly the same distance to walk
to work. Now where could they live?
y
8. After a day of fruitless apartment hunting, they
decide to widen their area of search. The only
requirement that they keep is that they both be the
same distance from their jobs. Now where should
y
they look?
x
A)
B)
C)
D)
x
True for Euclidean geometry
True for Taxicab geometry
True for both Euclidean geometry and Taxicab geometry
False for both Euclidean geometry and Taxicab geometry
Choose the best answer from the above list for each.
_________9. Every segment has a midpoint.
________10. The distance between two points is a constant.
________11. A circle is a continuous set of points.
________12. A circle is a finite set of points.
________13. Only integers can be coordinates of a point.
________14. All points on the plane have coordinates. or
Every point on the plane has coordinates.
________15. Exactly one segment can be drawn between any two points.
________16. Points on a perpendicular bisector of a segment are equidistant from the endpoints of the
segment.
________17. Not all segments have perpendicular bisectors.
________18. A perpendicular bisector is an infinite set of points.
________19. A perpendicular bisector is a continuous set of points.
________20. A perpendicular bisector is a discrete set of points.
________21. A circle is a discrete set of points.
________22. Exactly one segment is the shortest distance between two points.
THursday
Euclidean/Non-Euclidean Geometry
About two thousand years ago, Euclid summarized the geometric knowledge of his day. He developed this
geometry based upon ten postulates. The wording of one of his postulates, known as the parallel postulate,
was very awkward and received much attention from mathematicians. These mathematicians worked
diligently to prove that the conclusions in Euclidean geometry were independent of this parallel postulate. A
mathematician named Saccheri wrote a book called Euclid Freed of Every Flaw in 1733. His attempt to show
that the parallel postulate was not needed actually laid the foundation for the development of the two
branches of non-Euclidean geometry. Euclidean geometry assumes that there is exactly one line parallel
to a given line through a point not on that line. The branch of non-Euclidean geometry called spherical
or Riemannian assumes that there are no lines parallel to a given line through a point not on that line.
The other branch of non-Euclidean geometry called hyperbolic or Lobachevskian geometry assumes that
there is more than one line parallel to a given line through a point not on that line.
Physical models for these geometries allow us to visualize some of their differences. The model for
Euclidean geometry is the flat plane. The model for hyperbolic geometry is the outside bell of a
trumpet. The model for spherical geometry is the sphere.
I.
We have proved that the sum of the angles of a triangle is 180. On a globe, is it possible to have a
triangle with more than one right angle? _________ Is this a Euclidean triangle? _______ Why or
why not? _______________________________________
The sides of this triangle (on the globe) curve through a third dimension. The surface upon which the
triangle is drawn affects the conclusions about the sum of its angles. Euclidean geometry is true for
measurement over relatively short distances (when the surface of the earth approximates a flat plane).
Remember the physical experiences possible when this geometry was developed. The geometry of
Einstein’s theory of relativity is the geometry of no parallel lines (spherical or Riemannian). Notice that
these non-Euclidean geometries are derived from different postulates.
II. A second type of non-Euclidean geometry results when a single definition
is changed. Euclidean geometry defines distance “as the crow flies.” In
other words, distance is the length of the segment determined by the
two points. However, travel on the surface of the earth (the real world)
rarely follows this ideal straight path.
y
On the grid at the right, locate point A with coordinates (-4, -3) and
point B with coordinates (2, 1). Use the Pythagorean Theorem to find
the Euclidean distance between A and B.
Now consider that the only paths that can be traveled are along grid
lines. This distance is called the “taxi-distance.” What is this “taxidistance” from A to B?
Points on a taxicab grid can only be located at the intersections of horizontal and vertical lines.
One unit will be one grid unit.
Therefore, the numerical coordinates of points in taxicab geometry must always be _____________.
The taxi-distance between 2 points is the smallest number of grid units that an imaginary taxi must
travel to get from one point to another.
1. Two points determine a line segment. (a segment is the shortest distance between two points)
(a) Draw a taxi segment from point A to point B. What is the length of this segment?
B
_________





x
(b) Is this the only taxi segment between the two points? ______
If not, how many different taxi segments can you draw between points A and
B? _______
(c) In taxicab geometry, do two points determine a unique segment? ________
2. A circle is the set of points in a plane that are the same distance from a given point in the plane.
(a) On the grid at the right, draw a taxi-circle with center P and a
radius of 6.
(b) Is this the only taxi-circle that can be drawn with this center and
this radius? ______ If not, how many different taxi-circles can be
drawn? _______
P

(c) Can you draw a Euclidean circle without lifting your pencil?
_______ ; the “taxi-circle”? _____ The “taxi-circle” is an example
of discrete mathematics where the sample space is a set of
individual points (not a continuous set such as a number line).
3. A midpoint, M, of a segment, AB , is a point on the segment such that AM = MB.
Q
(a) Find the midpoint of segment PQ.
(b) Is there more than one midpoint? ______
(c)
Find the midpoint of segment PT.
T
(d) What conclusion can you make about the number of midpoints in taxicab
geometry?
P
4. A point is on a segment’s perpendicular bisector if and only if it is the same distance from each of
the segment’s endpoints.
(a) Find all points that satisfy
(b) Find all points that satisfy
(c) Find all points that satisfy
this definition in taxicab
this definition in taxicab
this definition in taxicab
geometry for segment DE.
geometry for segment ST.
geometry for segment KL.
S

D

L

E
T
(c)

K
What conclusion can you make about perpendicular bisectors in taxicab geometry?
Comparing Planar and Spherical Geometry
Complete the table below to compare and contrast lines in the system of plane Euclidean geometry and
lines (great circles) in spherical geometry.
On the plane
On the sphere
1. Is the length of a line finite or
infinite?
2. Describe the shortest path
that connects two points.
3. Can you extend a line
forever?
4. How many parts (and are
they finite or infinite) will two
points divide a line?
5. How many lines pass through
any two different points?
6. How many lines are parallel
to a given line and pass through
a given point not on the given
line?
7. If three points are collinear,
exactly one is between the other
two. (True or false)
For each property listed from plane Euclidean geometry, write a corresponding statement for spherical
geometry.
8.
Two distinct lines with no point of intersection are parallel.
9.
Two distinct intersecting lines intersect in exactly one point.
10. A pair of perpendicular lines divides the plane into four infinite regions.
11. A pair of perpendicular lines intersects once and creates four right angles.
12. Parallel lines have infinitely many common perpendicular lines.
13. There is only one distance that can be measured between two points.
14. There is exactly one line passing through two points.
OVER
Choose one of the following answers for each question: A)
B)
C)
D)
true on a plane
true on a sphere
true on both a plane and a sphere
true on neither a plane or a sphere
_____15. A line is an infinite set of points.
_____16. A line is continuous (no “holes” or gaps).
_____17. Through any two points, there is exactly one line.
_____18. There exists at least one pair of points through which more than one line can be drawn.
_____19. A polygon may have two sides.
_____20. Each angle of an equilateral triangle must be 60˚.
_____21. Each angle of an equilateral triangle may be 45˚.
_____22. Each angle of an equilateral triangle may be 120˚.
_____23. A line is bounded. (that is, it can fit into a closed box)
_____24. There is no greatest distance between two points.
_____25. Two lines can share no points.
_____26. Two distinct lines can share two points.
_____27. Two distinct lines can share more than two points.
_____28. The sum of the angles of a triangle is always the same number.
_____29. A triangle can have at most one right angle.
_____30. Three lines may be perpendicular to each other (that is, line a  line b  line c  line a)
_____31. Three lines may be parallel to each other.
_____32. Three lines may intersect in three points. (each of the lines intersects the other two lines)
_____33. Three lines may intersect in two points. (each of the lines intersects the other two lines)
_____34. Three lines may intersect in four points.
_____35. Vertical angles are congruent.
Ws 1: Geometry
Coordinates in Space
Name________________________________
Date___________________Period_________
1. On which axis does each of
the following points lie?
a) (5,0,0)
x
y
axis axis
z
axis
2. On which plane does each
of the following points lie?
a) (0,4,6)
b) (0,0,-2)
b) (-2,-1,0)
c) (0,3,0)
c) (3,0,-5)
d) (0,0,0)
d) (0,-2,3)
3. Write an equation for each
of the following planes:
XY
YZ
XZ
plane plane plane
a) XY plane
b) YZ plane
c) XZ plane
Match each of the following to a description of its graph. (include all descriptions that apply)
4. (3,0,0) __________
A. point on the X-axis
5. (0,0,0) __________
B. point on the Y-axis
6. (0,-6,0) __________
C. point on the Z-axis
7. (0,0,20) __________
D. point on the XY plane
8. (2,3,-1) __________
E. point on the YZ plane
9. (2,4,0) __________
F. point on the XZ plane
10. (-1,0,-1) __________
G. plane  to XY plane
11. (0,0,4) __________
H. plane  to YZ plane
12. x = -2 __________
I. plane  to XZ plane
13. y = 6 __________
J. plane
to XY plane
14. z = 0 __________
K. plane
to YZ plane
15. z = 7 __________
L. plane
to XZ plane
16. x = 0 __________
M. XY plane
17. 2x+3y=6 __________
N. YZ plane
18. 4x-2y=8 __________
O. XZ plane
19. 2x+5z=10 __________
P. Point in space
20. 7y-2z=14 __________
Q. Plane
to x-axis
R. Plane
to y-axis
S. Plane
to z-axis
21. Name three points on the graph of each:
(a) 3x – 2y + 4z = 12
( ___, ____, ____)
(b) 7x + 4y – 14z = 28
(c) 3x – 2y – 5z = 15
(d) x + y + z = 0
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
( ___, ____, ____)
22. Five of the eight vertices of a cube are points:
A(-1,3,-2), B(4,3,-2), C(4,-2,-2), D(-1,-2,-2), and E(4,3,3). Find coordinates for the other three
vertices.
( ___, ____, ____); ( ___, ____, ____); ( ___, ____, ____)
23. Five of the vertices of a rectangular solid are points:
A(-1,-1,-5), B(-1,-1,2), C(-1,3,2), D(-1,3,-5), and E(1,-1,-5). Find the coordinates of the other three
vertices.
( ___, ____, ____); ( ___, ____, ____); ( ___, ____, ____)
Thursday
5/22. Geometry
Worksheet 2 – Coordinates in Space
Name_____________________________
Date___________________Period______
Determine the distance between each pair of points, and determine the coordinates of the midpoint of the
segment connecting them.
1. C(4, -8, 12) and D(7, 20, 18)
2. E(3, 7, -1) and F(5, 7, 2)
3. G(2, 2, 2) and H(-25, 4, 18)
Identify each of the following as true or false. If the statement is false, explain why.
4. Every point on the yz-plane has
coordinates (c, y, z) for any real
number c.
5. The point at (1, 8, -12) is inside
the sphere
(x  3)2  (y  5)2  (z  2)2  9 .
7. The set of points in space 5 units from the point
at (1, -1, 3) can be described by the equation:
(x  1)2  (y  1)2  (z  3)2  25 .
6. The intersection of the xyplane, the yz-plane, and the xzplane is the point (0, 0, 0).
8. The set of points equidistant from A(2, 5, 8) and
B(-3, 4, 7) is a line that is the perpendicular bisector
of AB .
Determine the coordinates of the center and the measure of the radius for each sphere whose equation is given.
9. x2  (y  3)2  (z  8)2  81
10. (x  5)2  (y  4)2  (z  10)2  9
11. x2  y2  (z  3)2  49
12. (x  4)2  (y  2)2  (z  12)2  18
Write the equation of the sphere using the given information.
13. The center is at (-5, 11, -3),
and the radius is 4.
14. The center is at (-2, 3, -4) and
it contains the point at (5, -1, -1).
16. It is concentric with the sphere with equation
(x  5)2  (y  4)2  (z  19)2  9 , and it has a radius of
6 units.
15. The diameter has endpoints at
(14, -8, 32) and (-12, 10, 12).
17. It is inscribed in a cube determined by the
points at (0, 0, 0), (4, 0, 0), (0, 4, 0), and (4, 4, 4).
18. Find the perimeter of a triangle with vertices A(-1, 3, 2), B(0, 2, 4), and C(-2, 0, 3).
19. Show that ∆ABC is an isosceles right triangle if the vertices are A(3, 2, -3), B(5, 8, 6), and C(-3, -5, 3).
20. Consider R(6, 1, 3), S(4, 5, 5), and T(2, 3, 1).
(a) Determine the measures of RS,ST, and RT .
(b) If RS,ST, and RT are sides of a triangle, what type of triangle is ∆RST?
21. Find the surface area and volume of the rectangular prism at the right.
22. Find z if the distance between R(5, 4, -1) and S(3, -2, z) is 7.