Download (Points, Lines, Planes and Transformations)

Geometry: Points, Lines, Planes: Do Now
State whether each is best modeled by a point, line, or plane.
1. a knot in a piece of thread
2. a piece of cloth
3. the corner of a room
4. the telecommunications beam to a satellite in space
5. the crease in a folded sheet of wrapping paper
6. an ice skating rink
List two to three real-world objects that could be modeled using a
point, a line, and a plane.
Name:
Date:
Geometry: Points, Lines, Planes: Notes
Points, Lines, and Planes
Point
A
 Undefined
 A point has no size, no dimension.
 It is a location in space.
 Points are named using capital letters.
Line
AB
l
 Undefined
 A line is made up of an infinite number of points that go on
without end in both directions.
 A line has one dimension, length.
 A line can be named with a single lowercase script letter or by
two points on the line.
Ray
AB
 Part of a line
 A ray has a definite starting point and extends without end in
one direction.
 A ray is named using the endpoint first, then another point on
the ray.
Line Segment
AB
 A line segment is part of a line containing two endpoints and all
points between them.
 A line segment is named using the endpoints.
Collinear
 Points that lie on the same line.
Non-collinear
 Points that do not lie on the same line.
Plane
 Undefined
 A flat surface that extends without end in all directions.
 It has two dimensions, length and width.
 A plane can be named by three non-collinear points or by a
single uppercase script letter.
 A parallelogram is used to model a plane.
Coplanar
 Points that lie in the same plane.
Non-coplanar
 Points that do not lie in the same plane.
Space
 Boundless, three-dimensional set of all points
Parallel lines
 Lines in the same plane that are always the same distance
apart.
 They do not intersect.
Skew lines
 Lines that are not in the same plane and do not intersect.
Intersection
 The set of points that are in both figures.
Concurrent lines
 Three or more lines that pass through the same point.
Postulates: (Statements that are accepted as true without proof).
 Two points determine a unique line.
 If two distinct lines intersect, then their intersection is a point.
 Three non-collinear points determine a unique plane.
 If two distinct planes intersect, then their intersection is a line.
Important words: contains, determine, on, in, a, distinct, can, will.
Name:
Date:
Geometry: Points, Lines, Planes: Classwork
True or False
1.
Two points determine two lines.
2.
Two planes always intersect in a line.
3.
If two distinct lines intersect, they always intersect at a point.
4.
Three points determine a plane.
5.
If two planes intersect in a line, then the line is in both planes.
6.
Two planes can intersect in a point.
7.
It is possible for two lines to lie in the same plane.
8.
Three planes can intersect in a point.
9.
Two lines are parallel if they do not intersect.
10. KM and MK name the same set of points.
Multiple Choice
1. The intersection of 2 line segments can be
I. a point
III. a ray
A. I only
above
II. a line segment
IV. a line
B. I and II
C. I and IV
D. all of the
2. The intersection of 2 distinct rays can be
I.
a point
III. a ray
A. I only
II. a line segment
IV. a line
B. IV only
C. I, II and III
D. all of the above
3. The intersection of 2 distinct lines can be
I.
a point
III. a ray
A. I only
above
II. a line segment
IV. a line
B. IV only
C. I, III and IV
D. all of the
Questions:
1. What is the difference between:
a line and a ray?
a line and a line segment?
2. How many points are on a line?
3. Can a plane and a line ever intersect in two points?
4. If points A, B, & C determine plane D, what do you know about points A, B, &
C?
Draw:
1. Line segment AB
2. Line l and point P such that P is not on line l
3. Points A, B, & C such that they are collinear.
4. Lines r and t such that they have no points in common.
5. Points A, B, & C such that they determine a plane.
r
P is on r
r contains P
r passes through P
P
m
l and R are in N
N contains R and l
m intersects N at R
R is the intersection of m with
R
1. Do l and m intersect?
2. Does N contain m?
3. l and m are called ________
N
l
N
lines.
Draw:
1. Plane Q with a line m intersecting Q at point E.
2. Plane Q contains lines r and s that intersect in point P.
3. Line t lies in planes P, Q and R.
Name:
Date:
Geometry: Points, Lines, Planes: Exit Card
List one or two real-world objects that could be modeled using:
a segment
a ray
parallel lines
perpendicular lines
intersecting lines
concurrent lines
skew lines
intersecting planes
Name:
Date:
Geometry: Points, Lines, Planes: Notes
F
l
H
A
P
D
E
C
B
G
I
R
1. Points C, F, and E
A. determine a plane
C. are contained in only plane P
B. are collinear
D. are non-coplanar points.
2. The line that contains H and G
A. intersects plane P but not plane R.
plane P.
C. does not intersect either plane P or plane R.
plane R.
B. intersects plane R but not
D. intersects both plane P and
3. What is the intersection of planes P and R.
A. points A, D and B
C. AB
4. Line l and AB can be described as:
A. skew
C. parallel
B. AB
D. plane A, D, B
B. intersecting
D. co-planar
5. Write another name for plane P. _________________
6. Name the intersection of line l and plane R. _________________
True or False
7. Two planes can intersect in two lines. __________
8. If two lines intersect then only one plane contains both theOVER
lines. __________
9. AB and BA name the same set of points. __________
Draw:
10. Plane Z contains lines r and s that intersect in point P. Line t is skew to line r and
intersects line s in point Q.
Name:
Date:
Geometry: Points, Lines, Planes: Project
Mathematics Project:
Due Date:
Late Fee:
Point Value:
Grading Criteria:
Modeling Points, Lines and Planes
One letter grade drop for each day late.
40 points
1. Neat and clear drawing or model.
2. Points, lines and planes represented.
3. Demonstrates spatial relationships.
4. Proper notation used.
5. Probing questions and accurate answers.
Description of Project: Draw or make a model that represents the basic elements of
geometry – points, lines and planes. (You may also include rays and segments). Your
model must illustrate the following spatial relationships: parallel, skew, intersecting,
collinear, coplanar, non-collinear, and non-coplanar. You will probably need at least 2
planes, 3 lines and 5 points.
Write at least 4 questions that ask about specific elements and relationships in your
model. They can range from basic to complex. Include answers to your questions on a
separate piece of paper.
Geometry
Lesson Plan 2
Unit Title: Basics of Geometry
period
Time Estimate: 1
Lesson Title: Angle Relationships
Behavioral Objectives

The student will demonstrate analysis of types of angles by defining, identifying and
categorizing supplementary, complementary, adjacent, linear pairs, and vertical
angles.

The student will demonstrate synthesis by reaching conclusions about pairs of angles
and solving algebraic problems to find measurement.
Materials


Patty Paper
Slates
Lesson Procedures
Motivation:
Distribute patty paper. Fold to make line l. Fold to make line m such that it intersects
line l at point C. Label the angles clockwise 1, 2, 3, and 4. Ask students to write down
everything they can about the angles formed.
Activities:
1. Have students copy illustrations, definitions and angle relationships into their
notebooks: adjacent angles, vertical angles, linear pair, supplementary and
complementary.
2. Distribute slates. Display overhead. Work through questions 1 – 6.
3. Distribute classwork. Have students work in pairs, comparing answers and checking
notation for problems #1-6.
4. Page 2 of classwork. The teacher models the correct approach and solution to
algebraic problems in geometry as students copy the procedure. Continuing to work
in pairs, the students complete the algebraic problems as the teacher monitors their
work.
5. Distribute homework.
6. Additional Honors Geometry solving quadratics.
7. Optional SAT homework.
Closure:
Display sometimes, always, never statements one at a time. Have students draw an
illustration for each.
1. The supplement of an acute angle is acute.
2. Vertical angles are right angles.
3. An acute angle has a complement.
4. Supplementary angles are adjacent.
5. A complement is 90º.
Did you find the above sometimes, always, never questions challenging? Why or why
not?
Assessments:
 Class participation
 Slate responses
 Exit Question
 HSPA Open-Ended
 Homework
Geometry: Angle Relationships: Overhead
Complementary and Supplementary Angles
I. Definition – Two angles are complementary if
______________________________________
One angle is called the _____________ of the other.
1. Are the following 2 angles complementary? Answer yes or no.
a. 40º and 50º _____
b. 36º and 54º _____
c. 46º and 53º _____
2. Find the complement of the given angle:
a. 20º _____
c. 78º _____
b. 35º _____
d. xº
3. Are angles 1 and 2 complementary? Why?
a.
1 45º
2
45º
b.
1
2
Are complementary angles always adjacent?
II. Definition – Two angles are supplementary if
______________________________________ One angle is called the _____________
of the other.
4. Write 2 pairs of supplementary angles.
a. _____ and _____
b. _____ and _____
5. Find the supplement of the given angle:
a. 35º _____
b. 178º _____
6. Are angles 1 and 2 supplementary? Why?
c. 90º _____
d. xº
a.
b.
1 37º
1
2
2 143º
Name:
Date:
Geometry: Angle Relationships: Classwork
I. In the diagram, mAXB = 90º. Name
1. 2 congruent supplementary angles. ___________________
B
2. 2 supplementary angles that are not congruent. ________________
C
3. 2 complementary angles. _______________
4. a straight angle. ________________
A
X
D
II. Given, WE  CG
5. Name a pair of complementary angles. _______________
D
E
(Do not name the same pair as your partner).
F
6. Name a pair of supplementary angles. _______________
W
C
G
(Do not name the same pair as your partner).
H
Integrating Algebra into Geometric Concepts
Example Problem:
yº
3xº
5xº
Are the angles equal?
Do they add to 90º?
Solve.
Do they add to 180º?
Answer the question. Write the correct algebraic equation.
7.
8.
54º
4x
9.
3x
2x
11.
y
3x
75º
10.
(x + 10)º
(4x – 35)º
12.
(12x - 15)º
(3x + 45)º
(3x + 8)º
(5x – 20)º
(5x+4y)º
13. One angle is twice as large as its complement. Find the measure of both angles.
Name:
Date:
Geometry: Angle Relationships: Homework
J
1. Name the angle 4 ways.
a. _________
b. ________
c. ________
d. ________
T
1
M
2. Draw and label  KLM. Place point P on the interior of the angle, point Q on the
angle and point R on the exterior of the angle.
3. Identify each of the following pair of angles as complementary, supplementary or
vertical.
l1
1
1
2
1
l2
2
2
l1  l2
a. _________________
__________________
b. _______________
c.
4. In each of the following find the m2
OM  OL
M
47
2
2
155
38
2
O
a. ____________________
______________
L
b. ______________
c.
5. In each of the following, solve for x.
140
3x
4x +9
75
18
a. _x=_________
6. Given: OE  OC
12x-40
b. __x=_________
B C
D
c. __x=_________
A
O
E
a. Name a pair of complementary angles. ___________________
b. Name a pair of supplementary angles. ________________
c. Are BOC and COD complementary? (Support your answer) _____________
________________________________________________________________
7. Given: RT  RS and mTRW = 32.
Find m1.
T
W
1 R
S
8. The complement of the complement of an 80 angle is
a. 10
b. 80
c. 100
d. 280
9. Which of the following statements is always TRUE?
a. Vertical angles are supplementary
b. The complement of an acute angle is an obtuse angle.
c. The supplement of an obtuse angle is acute.
d. Complementary angles are congruent.
10. The measure of the angle made by the hands of a clock at 4:30 is:
a. 30
b. 45
c. 60
d. can’t be determined
E
D
F
Given, WE  CG and WD  WF and mDWE = 30º
C
W
G
11. Label the diagram with the measure of the other angles.
H
12. The measure of an angle is five times greater than its supplement. Find the measure
of the angle.
Name:
Date:
Honors Geometry: Angle Relationships: Homework
1. 1 and 2 are vertical angles. m1= x2 – 3 and m 2 = 6. Find x, m1 and m2.
2. 1 and 2 are complementary angles. m1=x2 + 60 and m 2=10x + 55. Find x,
m1 and m2.
3. 1 and 2 are vertical angles. m1= x2 + 5x and m 2 = x + 21. Find x, m1 and
m2.
4. 1 and 2 form a linear pair. m1= 3x2 + 100 and m 2 = x2 + 44. Find x, m1
and m2.
Name:
Date:
Geometry: Angle Relationships: Classwork
QUANTITATIVE COMPARISON
Answer: A if the quantity in column A is greater
B if the quantity in column B is greater
C if the quantities are equal
D if there is not enough information
Column A
Answer
Column B
1.
89
x y
x
Given:
2.
y
_____
x
_____
45
_____
180
_____
r+s
_____
25  144  x
13
(a, b)
3.
w
w
y
x
4.
x+y
x
y
5.
p
p+q
q
r
s
1. A triangle with one side 5 and another side 10 has perimeter, P. What are the least
and greatest possible integer values of P?
A. 21 and 29
B. 21 and 24
C. 16 and 39
D. 16 and 25
E. 6
and 29
2. Three lines intersect as shown in the figure above. What is the sum of the degree
measures of the marked angles?
A. 360
B. 540
C. 720
D. 900
E. cannot be
determined
3. In ABC below, what is the value of x?
A. 40
B. 42
C. 50
determined
B
D. 58
E. cannot be
x x
48
A
32
M
C
4. Five line segments intersect to form the figure below. If p + q + r = 285, what is the
value of t + s?
A. 60
B. 75
C. 90
D. 105
E. cannot be
determined
t
s
p
q
r
5. The side of a square is s, each side of a triangle is t. If their perimeters are equal,
express s in terms of t.
A.
4t
3
B.
3s
4
C. 4t = 3s
D. 4s + 3t
E.
3t
4
Name:
Date:
Geometry: Angle Relationships: Classwork
Quantitative Comparison
Answer: A if the quantity in column A is greater than the quantity in
column B
B if the quantity in column B is greater than the quantity in
column A
C if the quantities are equal
D if there is not enough information
Note: The figures are not drawn to scale
Column A
Answer
1.
Column B
x + 40
_________
110
x
1
y
y
 1 || 2
2.
2
a
_________
b
1
5a
3b
 1 || 2
3.
2
_________
a
b
x
1
a
b
2
x > 90
4.
The area of parallelogram ABCD
_________
B
10
5
C
2
75
A
Multiple Choice Questions
5. In the figure, if  1 || 2 , then p – q =
A. 40
B. 70
D
40
C. 100
D. 110
E. 140
q
p
6. In the figure  || m . Which of the following pairs of angles must be equal?
A. 1 and 2
1
B. 2 and 3
C. 3 and 5
2
3

D. 4 and 6
4
5
E. 5 and 7
6
7
m
7. In the figure, if PQ || OR , what is the area of quadrilateral PQRO?
A. 9
B. 14
P(0,4)
Q
C. 18
D. 36
45
E. Cannot be determined
O
R (5, 0)
8. In the figure above, if  1 || 2 , what is the value of x in terms of a?
A. 180 – a
B. 180 + a
a
C. 270 – a
D. 360 + a
x
1
E. 360 - a
2
9. If parallel lines  1 and  2 are intersected by  3 , what is the value of x + y?
A. 180
x
B. 150
C. 120
y
D. 90
E. Cannot be determined
10. In the adjoining diagram, if lines  , m, and n are parallel, then mx =
A. 50
B. 30
50

C. 20
m
D. 60
x
150
E. Cannot be determined
n
11. In the adjoining diagram, if lines  is parallel to line k, and line m is parallel to line
n. Find mx
A. 10
k
m
n

B. 20
C. 30
x
D. 40
100
110
E. 50
12. In the adjoining diagram, lines AB and CD are parallel. Express the measure of x
in terms of a and b. A. x = 180 – ab
B.
C.
D.
E.
x = 2(a + b)
x=b–a
x=a–b
x=a+b
A
a
B
x
b
C
D
13. In the adjoining diagram, the rays of x are cut by two parallel lines. Express the
measure of x in terms of a and b.
a
A. a – b
B. b – a
C. 180 – a + b
D. 180 – a – b
E. 180 – b + a
b
x
Geometry
Lesson Plan 3
Unit Title: Basics of Geometry
period
Time Estimate: 1
Lesson Title: Transformations
Behavioral Objectives

Comprehension of transformations (reflections, rotations, translations and glide
reflections) by defining and identifying mappings and isometries in terms of image
and pre-image.

Evaluation of transformations by discovering patterns on the coordinate plane
(reflections over the y = x, vertical and horizontal lines; translations up and down).
Materials




Laptops
Projector
Graph Paper
Class set of “F” that fit graph paper
Lesson Procedures
Motivation:
Show 2 minute video http://www.learnalberta.ca/content/mejhm/html/video_interactives/transformations/transf
ormationsSmall.html
Activities:
1. Since transformations are about movement, each student is given a shape to move.
Distribute graph paper and the letter “F” or other non-symmetric letter. Ask students
to place the “F” in Q1. Ask them to rotate the “F”. Quickly students will ask “how”,
“which way”, etc. Display table with answers covered on Smartboard. Ask students
to think about what directions are needed in order to have everyone rotate the “F” the
same way. Similarly, discuss translation and reflection. Display/write answers in
table.
2. Display website: http://mathforum.org/sum95/suzanne/symsusan.html Have
students copy the definition of Glide Reflection. Use the illustrations to define and
illustrate the terms image, pre-image, mapping and isometry. Next display
http://illuminations.nctm.org/LessonDetail.aspx?id=L475 have students answer
questions 1-10.
3. Students, working on laptops go to the following site:
http://www.learnalberta.ca/content/mejhm/html/video_interactives/transformations/tr
ansformationsInteractive.html Students complete 10 problems. For each problem,
students write down the coordinates of the pre-image and the image of the purple star.
Projector option: Group 3-4 students together according to ability. Have groups
decide on correct response. Give the cordless mouse to a group and have them select
their answer.
4. Working in groups of three students complete the classwork. As groups determine
rules for reflections, translations, and rotations they are written on the board. The
teacher confirms the correct rule to be written in notebooks.
5. Distribute homework.
Closure:
Assign exit ticket and collect.
Assessments:
 Class participation
 Completion of Computer Actitivity
 Completion of Group Worksheet
 Exit Ticket
 Homework, including HSPA multiple choice questions
Geometry: Transformations: Overhead
Transformation
Rotation
Common term
Directions
Common
term
Spin
Directions
Translation
Reflection
Glide Reflection
ANSWERS (move shape)
Transformation
Rotation – A transformation in
which every point moves along a
circular path around a fixed point
Translation – A transformation
that shifts or moves every point the
same distance and the same
direction.
Reflection – A transformation in
which every point is the same
distance from the line of reflection
or axis of symmetry.
Slide
Flip /
Mirror
Image
About a point (i.e. the origin, the
center)
Number of degrees
Clockwise or counterclockwise
Direction (i.e. up/down, left/right)
Number of units
Over a line (i.e. x-axis, y-axis, y=x)
Name:
Date:
Geometry: Transformations: Classwork
1. Graph ∆TOP with coordinates T(-3, 1), O (5, 3) and P(0, 6).
a. Reflect ∆TOP over the x-axis.
Pre-Image
Image
T(-3, 1) →
O (5, 3) →
P(0, 6)
→
b. Reflect ∆CAN over the y-axis.
Pre-Image
Image
C (-6, 3) →
A (-2, -4) →
N (0, 7) →
2. Use the above exercise to write a general rule for reflections.
a. The image of (a, b) reflected over
the x-axis would be _________.
b. The image of (a, b) reflected over
the y-axis would be _________.
3. Translate ∆CAN 2 units to the right and 6 unit down, written 2,6
Pre-Image
Image
C (-6, 3) →
A (-2, -4) →
N (0, 7) →
4. Use the above exercise to write a general rule for translations.
The image of (a, b) translated h, k would be __________.
5. Rotate ∆TOP 180º about the origin.
Pre-Image
Image
T(-3, 1) →
O (5, 3) →
P(0, 6)
→
6. Use the above exercise to write a general rule.
The image of (a, b) rotated180º about the origin would be ________.
Bonus:
________.
The image of (a, b) rotated 90º clockwise about the origin would be
The image of (a, b) rotated 90º counterclockwise about the origin would be ________.
Name:
Date:
Geometry: Transformations: Exit Problem
For the following diagrams, name the transformation and the specific instructions such
that the pre-image is mapped onto the image.
1. Transformation _______________
Instructions
_________________________________
2. Transformation _______________
Instructions
_________________________________
3. Transformation _______________
Instructions
_________________________________
4. State the coordinates of the image given the following transformation of point P(4, 2):
a.
b.
c.
d.
reflected over the y-axis _________________.
translated  9,3 _________________.
rotated 180º about the origin ________________.
rotated 90º counterclockwise about the origin.
Name:
Date:
Geometry: Transformations: Homework
1. A reflection (flip) is one form of a transformation. In the Video Interactive shapes are
only reflected over the ____ or ____ axis.
2. a) This shape has been reflected over the ____ axis.
b) This shape has been reflected over the
____ axis.
3. A translation (slide) is also a form of a transformation. Shapes may be translated by
moving them either _____________ or ________________.
4. a) This shape has been translated ____ units _________, and ___ units ________.
(Left/Right)
(Up/Down)
5. a) This shape has been translated ____ units _________, and ___ units ________.
(Left/Right)
(Up/Down)
6. A rotation (turn) is another form of a transformation.
Choose a matching rotation for the following:
a) What is the same as a 90 clockwise rotation? ___________________________
b) What is the same as a 180 clockwise rotation? ___________________________
c) What is the same as a 270 clockwise rotation? __________________________
7. a) This shape has been rotated ____ degrees
clockwise or ____ degrees counterclockwise.
b) This shape has been rotated ____ degrees
clockwise or ____ degrees
counterclockwise.
8. State the coordinates of the image given the following transformation of point P(-5,
4):
a. reflected over the x-axis _________________.
b. translated  3,5 _________________.
c. rotated 180º about the origin ________________.
9. Find some patterns in your environment that were created using glide reflections.
Make a sketch of the pattern.
OR
9. Use glide reflections to create your own patterns.
10. Translate ∆ABC, 7 units up and 2 units right.
Reflect the translated image over the y-axis.
Draw ∆A’B’C’
A
B
C
HSPA/SAT Multiple Choice Problems:
R
R
11. Which figure could be a translation of R?
A.
B.
C. R
D. Я
12. What are the coordinates of the image of (5, 0) under a rotation of 90º clockwise
about the origin?
A. (-5,0)
B(0,5)
C. (0,-5)
D. (5,5)
13. If ∆ABC with vertices A(-2,0), B(-2,-2), C(-4,-2) is reflected over the x-axis and then
the image is reflected over the y-axis, which coordinates would represent the final image
of A?
A. (-2,0)
B. (2, 0)
C. (0, 2)
D. (0, -2)
Geometry
Lesson Plan 4
Unit Title: Basics of Geometry
period
Time Estimate: 1
Lesson Title: Transformations
Behavioral Objectives

The student will demonstrate synthesis of transformations by using computer
software to make and verify conjectures about geometric transformations.
Materials


Computer Lab
http://homepage.mac.com/dscher/river.html (teacher resource)
Lesson Procedures
Motivation:
There are many classic math problems, such as, the birthday problem, painted cube,
squares on a checkerboard, etc. Today you will investigate and solve the “burning tent”
problem. Pose the following problem: A camper out for a hike is returning to her
campsite. The shortest distance between her and her campsite is along a straight line, but
as she approaches her campsite, she sees that her tent is on fire! She must run to the river
to fill her canteen, then run to her tent to put out the fire. What’s the shortest path she can
take?
Activities:
1. Divide students into differentiated groups according to ability. Distribute classwork.
2. Pose the following exit question: Explain how this activity relates to the geometry of
billiards or pool.
3. Assign homework. The open-ended comes from the state website. A follow-up
activity could be designed in which the students rank the answers provided on state
site. http://www.state.nj.us/education/assessment/hs/hspa_mathhb.pdf
Closure:
Assign exit question and collect.
Assessments:
 Class participation
 Completion of classwork
 Exit Ticket
 HSPA Open-Ended (Homework)
Name:
Date:
Geometry: Transformations: Classwork A
A camper out for a hike is returning to her campsite. The shortest distance between her
and her campsite is along a straight line, but as she approaches her campsite, she sees that
her tent is on fire! She must run to the river to fill her canteen, then run to her tent to put
out the fire. What’s the shortest path she can take?
1. Make a pencil illustration of the problem. (Have the teacher approve the drawing
before going on to the next step).
2. Create a sketchpad illustration.
3. Determine the length of various paths (camper to river to tent).
4. Propose a method for finding the exact point on the river in order to minimize the
path.
Name:
Date:
Geometry: Transformations: Classwork B
A camper out for a hike is returning to her campsite. The shortest distance between her
and her campsite is along a straight line, but as she approaches her campsite, she sees that
her tent is on fire! She must run to the river to fill her canteen, then run to her tent to put
out the fire. What’s the shortest path she can take?
1. Make a pencil illustration of the problem. (Have the teacher approve the drawing
before going on to the next step).
2. Create a sketchpad illustration.
3. Determine the length of various paths (camper to river to tent).
4. What is true about the incoming and outgoing angles when the path is the shortest?
5. Use the idea of reflections to reflect a point about a line. Consider the fact that the
shortest distance between two points is a straight line. Propose a method for finding the
exact point on the river in order to minimize the path.
Name:
Date:
Geometry: Transformations: Classwork C
A camper out for a hike is returning to her campsite. The shortest distance between her
and her campsite is along a straight line, but as she approaches her campsite, she sees that
her tent is on fire! She must run to the river to fill her canteen, then run to her tent to put
out the fire. What’s the shortest path she can take?
1. Construct a long horizontal segment AB to represent the river. Then construct points
C and D on the same side of the segment (Point C represents the camper and D represents
the tent). Place a point E anywhere on AB and then construct the segments CE and DE to
represent the camper’s path.
Measure the lengths of CE and DE using the Measure Menu. Then, use the calculator to
find the sum of these two lengths. Finally, measure angle CEA and angle DEB (be
careful to select the points in the angle in the correct order).
2. Drag point E and watch the calculated sum change. Move point E to a location on the
river to which the camper should run. What appears to be true about the incoming and
outgoing angles?
3. So far you have dragged point E to find an approximate minimal path. Now let’s see
how to construct such a path. First, double click on segment AB to mark it as a mirror.
Which kind of transformation uses a mirror?
4. Select point D and choose Reflect from the Transform menu to create point D’.
Construct segment CD’ and change its line width to dashed in the Display Menu.
Measure the length of CD’. What do you notice about the length of CD’?
5. Where should point E be located in relation to CD’ and AB so that the sum CE + ED is
minimized?
Name:
Geometry: Transformations: Homework
Date: