Download Aim: How do we determine a probable locus

Day 1: Aim: How do we determine a probable locus?
Objectives: SWBAT
 State the definition of locus
 State and illustrate the five fundamental locus theorems
 Describe each locus in a complete sentence
Do Now: Describe the location of our school using as much detail as possible.
Mini Lesson:
Locus is a set of points which satisfies a certain condition. In Latin, the word locus
means place. Locus is all about visualizing location - where something is located and
making a detailed sketch is usually the best way to solve a locus problem.
When we think about a locus we think about a bunch of points all doing the same thing.
Theorem 1 - One Point - Students will act out this theorem.
All points on a circle are equidistant from the center of the circle. The distance of any
point from the center of the circle is the radius.
Students will be 4 ft from the center. If everyone is 4 ft away what path are your
classmates forming?
Equidistant - means equally distant of equally as far. All points on a circle are
equidistant from the center.
Theorem 2 - From one line
All the points equidistant from a given line is a parallel line.
Draw a line. Place the eyes on the paper so that they are the same distance from the line,
say 3 cm.
Theorem 3 - From two points
All points on the perpendicular bisector of a line segment connecting two points are
equidistant from the two points.
Students will use the geoboard to examine the distance from two points to the
perpendicular bisector
Theorem 4 - Two Lines
The locus of points equidistant from two given parallel lines is a parallel line midway
between them.
Students will draw two parallel lines and place the eyes such that they are equally as far
as the two lines.
Theorem 5 - Intersecting Lines
An angle bisector is equidistant from the sides of the angle
Students will draw and angle and then construct the angle bisector. They will then
measure the distance from the bisector to each ray. Have students then draw two
intersecting lines...4 angles.
Geometry/Period ___
Describing locus notes
Name: ________________________
Date: _________________________
Investigation: The 5 Locus Theorems
Locus: _________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Equidistant: ____________________________________________________________
 Theorem 1 - Distance from a Point
Sketch - What does the locus of points Description - What path do the points
look like?
form?
 Theorem 2 - Distance from a Given Line
Draw a line. Place the manipulatives on Description - What path do the points
the paper so that they are 3 cm from the form? What is the relationship of the
line. Remove the manipulatives and manipulatives to the given line?
sketch the result.
 Theorem 3 - Distance from two given points
Sketch
Description - What line can be
constructed that is equally distant
between two points?
 Theorem 4 - Distance from two parallel lines
Draw two parallel lines.
Place the Description - What is the relationship
manipulatives so they are ½ way between the locus formed by the
between both lines.
manipulatives and the given lines?
 Theorem 5 - An Angle
Draw an angle, and construct its Description - Measure the distance from
bisector.
the bisector to the ray, what do you
notice about this relationship?
Geometry/Period ___
Describing locus alternative stations activity
Name: ________________________
Date: _________________________
Station 1
Locus of Points Equidistant from a Point
●
Description of what your locus looks like as compared to the given point.
How would we be able to construct this locus using a compass and straightedge?
Name____________________________
Station 2
Locus of points equidistant from 2 points
●
●
Description of what your locus looks like compared to the two given points.
How would we be able to construct this locus using a compass and straightedge?
Name_____________________________
Station 3
Locus of points equidistant from two lines
Description of what your locus looks like compared to the two given lines.
How would we be able to construct this locus using a compass and straightedge?
Name_________________________________
Station 4
Locus of points equidistant from one line
Description of what your locus looks like compared to what the given line is.
How would we be able to construct this locus using a compass and straightedge?
Name__________________________________
Station 5
Locus of points equidistant from two intersecting lines
Description of what your locus looks like compared to what the intersecting lines look
like.
How would we be able to construct this locus using a compass and straightedge?
Geometry/Period ____
Describing locus hw
Name: _______________________
Date: ________________________
Directions: Think about each of the situations below. Describe the locus of points that
would satisfy the given criteria and sketch the probable locus.
1. The student radio station has a broadcasting range of 24 miles. Describe the locus
of points which represent the outer edge of the broadcasting range.
2. Describe the locus of points in a plane that are equidistant from two parallel lines.
3. Describe the locus of points in a plane that are a distance of 7mm from line r.
4. Two points are 6 cm apart. What is the locus of all points in a plane equidistant
from the two given points?
Turn Over! More on the back!
Regents Review!
1
In the accompanying diagram, line
1
is parallel to line
2
.
Which term describes the locus of all points that are equidistant from line
line 2 ?
(1) line
(3) point
(2) circle
(4) rectangle
2
1
and
The locus of points equidistant from two sides of an acute scalene triangle is
(1) an angle bisector (3) a median
(2) an altitude
(4) the third side
3
If point P lies on line
from point P?
, which diagram represents the locus of points 3 centimeters
(1)
(3)
(2)
(4)
Geometry/Period ___
Compound Loci notes
Name: _______________________
Date: ________________________
How do we solve problems involving Compound Loci?
A compound locus problem involves two, or possibly more, locus conditions occurring
at the same time. The different conditions in a compound locus problem are generally
separated by the word "AND" or the words "AND ALSO".
Steps for solving a compound locus problem
1. Prepare each condition separately on the SAME DIAGRAM. You many need to
create the diagram yourself!
2. After the two conditions are drawn, count the number of points where the two loci
conditions intersect.
3. The intersection is the point(s) that satisfy both conditions.
Example:
A treasure is buried in your backyard. The picture below shows your backyard which
contains a stump, a teepee, and a tree. The teepee is 8 feet from the stump and 18 feet
from the tree. The treasure is equidistant from the teepee and the tree and also 6 feet
from the stump. Locate all possible points of the buried treasure.
1. Maria’s backyard has two trees that are 40 feet apart, as shown in the
accompanying diagram. She wants to place lampposts so that the posts are 30 feet
from both of the trees. Draw a sketch to show where the lampposts could be
placed in relation to the trees. How many locations for the lampposts are possible?
2. The distance between parallel lines  and m is 12 units. Point A is on line  .
How many points are equidistant from lines  and m and 8 units from point A.
3. In the diagram below, town C lies on straight road p. Sketch the points that are 6
miles from town C. Then sketch the points that are 3 miles from road p. How
many points satisfy both conditions?
4. In the accompanying diagram, point P lies 3 centimeters from line . How many
points are both 2 centimeters from line  and 1 centimeter from point P?
Geometry/Period ___
Compound Loci hw
Name: _______________________
Date: ________________________
Compound Loci Homework Problem - 4 Points
A treasure map shows a treasure hidden in a park near a tree and a statue. The map
indicates that the tree and the statue are 10 feet apart. The treasure is buried 7 feet from
the base of the tree and also 5 feet from the base of the statue. How many places are
possible locations for the treasure to be buried? Draw a diagram of the treasure map, and
indicate with an X each possible location of the treasure.
Grading Rubric = Maximum Value is 4 points
[4]
A correct diagram is drawn, two X points are marked, a numerical 2 is
given for the places to dig, and appropriate work is shown.
[3]
The diagram is correct including two X points, but an incorrect answer or
no answer is found.
[2]
One correct locus situation and one incorrect locus situation are drawn, but
the answer is appropriate according to the diagram.
or
[2]
Each locus situation is correctly drawn, but no X points are marked, and no
numerical answer is found.
[1]
Only one locus situation is correctly drawn and an incorrect conclusion or
no conclusion is shown.
or
[1]
2 but no work is shown.
[0]
A zero response is completely incorrect, irrelevant, or incoherent or is a
correct response that was obtained by an obviously incorrect procedure.
Geometry/Period __
Compound Locus Day 2
Name: _________________
Date: __________________
Do Now:
What is the locus of all points in the plane 4 cm from a given line?
(1) a line perpendicular to the first
(2) two parallel lines 4 cm from either side of the first
(3) a parallel line 4 cm from the first
(4) a circle whose center is on the line
Locus Question Grading Rubric
[2] Both loci are sketched correctly, and the points of intersection are labeled with an X
[1] Both loci are sketched correctly, but the points of intersection are not labeled or are
labeled incorrectly.
or
[1] Appropriate work is shown, but one conceptual error is made, but appropriate points
of intersection are labeled.
[0] A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure
Compound Loci Classwork
1. The length of AB is 3 inches. On the diagram below, sketch the points that are
equidistant from A and B and sketch the points that are 2 inches from A. Label with an X
all points that satisfy both conditions.
2. Two lines AB and CD intersect at E. How many points are 2 units from E and also
equidistant from AB and CD ?
Geometry/Period ___
Name: _______________________
Date: ________________________
Compound Loci Treasure Hunt Portfolio Assignment
1. Ms. Terraferma buried a treasure chest filled of gold coins somewhere in New
York Harbor. The only clue she gave was that it was equidistant between the
Statue of Liberty and Governor’s Island as well as 4 miles from the Empire State
Building. Sketch all the possible locations Ms. Terraferma could have buried the
treasure. (Hint, use 1 cm = 1 mile on the compass)
2. Ahoy Mates! Pirate Pete is looking for the buried treasure. Unfortunately the
map he was given was labeled incorrectly. Captain Jack told him the buried
treasure is equidistant from Death Creek and Hangman’s Tree and also
equidistant from Smuggler’s Cove and Blue Lake. Can you find where the
treasure is really buried?
3. As part of WHSAD’s annual Spirit Week, Mr. Cornell buried a $100 McDonald’s
gift certificate near the school. That’s a lot of Happy Meals ! The only clue he
gave was that it was buried in a place equidistant from both Roebling Street and
Havemayer Street and that it was also equidistant between Metropolitan Avenue
and Meeker Avenue. Where did Mr. Cornell bury the gift card?
WHSAD
Writing for Understanding
4. Do you think this is a good location to bury the gift cards, why or why not?
5. If you were going to bury a treasure chest near school what clues would you give
to help students locate the chest?
6. What are the steps for solving a compound loci problem?
7. How is a compound loci problem like a buried treasure problem?
Grading Rubric – 20 points
Each Loci Question will be graded on a 4 point scale and each writing questions will
receive 2 points
[4] a correct diagram is drawn, the X points are marked and numerical answer is given,
and appropriate work is shown.
[3] The diagram is correct including the points of intersection, but the points of
intersection are not stated.
[2] One correct locus situation and one incorrect locus situation are drawn, but the answer
is appropriate according to the diagram.
or
[2] Each locus situation is correctly drawn, but no X points are marked and no numerical
answer is found.
[1] Only one locus situation is correctly drawn and an incorrect conclusion or no
conclusion is shown.
or
[1] Correct numerical answer is given, but no work is shown
[0] A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
Geometry/Period ___
Equations of linear loci
Name: ____________________
Date: _____________________
Write a description and the equation of the locus of points for each picture below.
Description:
Description:
Equation:
Equation:
Description:
Description:
Equation:
Equation:
Regent’s Practice
1. The locus of points equidistant from the points (4,-5) and (4,7)
is the line whose equation is
(1) y = 1
(3) x = l
(2) y = 2
(4) x = 4
2. Which equation represents the locus of all points 5 units below
the x-axis?
(1) x = -5
(3) y = -5
(2) x = 5
(4) y = 5
3. The equation y = 4 represents the locus of points that are
equidistant from which of the following two points?
(1) (4, 4) and (-4, 4)
(2) (0,0) and (0,8)
(3) (0,3) and (0,1)
(4) (4,0) and (0,4)
4. What is an equation of the locus of points 3 units below the
line whose equation is y = 2?
(1) y = -1
(3) y = 5
(2) x = 5
(4) x = -1
5. What equation describes a set of points equidistant from the xaxis and the y-axis?
(1) x = -7
(3) x = y
(2) y = 3
(4) x + y = 1
6. A point P(x,y) moves such that it is equidistant from the points
A(2, 3) and B(4, -3).What is the equation that represent this
locus? Graph is optional.