Download Homework on Loci

Geometry Semester 2
Homework on Locus Problems
Name: _______________________________
You will need a ruler and compass for this assignment.
1.
Draw a sketch of the locus described. If it is a line or lines, place a few points first and then
connect them to determine the locus. Then describe the locus as accurately as you can.
a.
The locus of points that are equidistant from the endpoints of segment AB:
A
b.
B
The locus of points that are 3 cm from A:
A
c.
The locus of points that are equidistant from the sides of angle A:
A
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d.
e.
The locus of points that are equidistant from lines AB and CD:
A
B
C
D
The locus of the points that are equidistant from lines MN and PQ:
Q
M
N
P
f. The locus of points P such that angle APB is a right angle:
A
B
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2.
Draw a diagram of the locus and write a description of the locus.
a.
Point P lies on line  . What is the locus of points on  and 3 cm from P?
b.
Point Q lies on line  . What is the locus of points 5 cm from Q and 3 cm from  ?
c.
Points A and B are 3 cm apart. What is the locus of points 3 cm from both A and B?
d.
Given triangle ABC, the locus of points that are equidistant from its sides consists of a
single point. What is that point called?
e.
Given triangle ABC, the locus of points that are equidistant from its vertices consists of a
single point. What is that point called?
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3.
Constructions with straightedge and compass are based on
locus problems. The purpose of this exercise is to analyze
the following construction problem:
Given circle O and segment AB, inscribe in the circle an
isosceles triangle PQR with base PQ the same length as AB.
a.
O
Suppose P has been chosen. Where must Q lie so that
PQ has the same length as AB? (In other words, what is
the locus of points that are AB units from P?)
A
B
b.
Now suppose segment PQ has been drawn. Where must R lie so that PR = QR? (In other
words, what is the locus of points equidistant from P and Q?)
c.
Explain the steps of the construction shown:
Step 1:
Step 2:
O
P
Step 3:
O
O
P
P
Q
d.
Q
Finish the construction:
O
P
Q
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4.
Two different solutions, both correct, are shown for the following construction problem.
Analyze the diagrams and explain the solutions.
Given segments with lengths r and s, construct triangle ABC with right angle C, and side
lengths AB = r, AC = s.
r
a.
s
First solution (start with A):
C
B
A
b.
Second solution (start with C):
A
C
B
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5.
The following locus is not a line or a circle:
Given point P and line , the locus of points that are the same distance from P as from line  is
called a parabola. P is called the focus and line  is called the directrix of the parabola.
Sketch a parabola with the line and directrix given below by following these steps:
a.
Locate the point V that is on the parabola and the line through the focus that is
perpendicular to the directrix. (V is called the vertex of the parabola.)
b.
Locate two points A and B that are on the parabola and the line through the focus that is
parallel to the directrix.
c.
Locate four points C, D, E, and F that are on the parabola and lines through the focus that
make 45o angles with the directrix. To make this accurate, consider the following diagram.
where PC = CQ. In that diagram the larger square’s sides have length equal to PC.
P
C
45o

d.
Q
R
Draw a curve through the 7 points you located.
P

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6.
Given segment AB, sketch the locus of points that are twice as far from B as they are from A.
This locus is not a line or a circle. First draw horizontal segment AB that is 6 cm long with A on
the left. Then follow these steps to draw the locus:
a.
There are two points of this locus on the line through A and B. Locate those points and
label them C and D, with C on the left and D between A and B.
b.
Construct the line perpendicular to AB at point A and locate two points, E and F, of the
locus that are on this line. (Hint: What kind of special triangle is triangle EAB?)
c.
Notice that AE  AF  2 3 because triangles EAB and FAB are 30-60-90 triangles.
d.
Locate points G and H on the intersections of the line perpendicular to AB and 4 cm left of
A with the lines parallel to AB through points E and F. These points are also on the locus,
since AG  AH  2 7 and BG  BH  4 7 :
2
c 2   2 3   4 2  12  16  28


E
G
c  28  2 7
d
c
2 3
 2
d 2  2 3  10 2  12  100  112
C
e.
4
A
6
B
d  112  4 7
Finally draw the closed curve containing those points. This curve is called an ellipse.
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7.
The following problem appeared in a book (One, Two, Three…Infinity) by a renowned scientist,
George Gamow:
A young man found among his great-grandfather's papers a piece of parchment that revealed
the location of a hidden treasure in a certain deserted island:
“There lieth a large meadow where standeth a lonely oak and a lonely pine. There thou wilt see
also an old gallows on which we once were wont to hang traitors. Start thou from the gallows
and walk to the oak counting thy steps. At the oak thou must turn right by a right angle and take
the same number of steps. Put there a spike in the ground. Now must thou return to the gallows
and walk to the pine counting thy steps. At the pine thou must turn left by a right angle and see
that thou takest the same number of steps, and put another spike into the ground. Dig halfway
between the spikes; the treasure is there.”
The young man found the deserted island, the meadow, the oak and the pine. But alas, the
gallows was long gone, no trace even of where it once stood!
To paraphrase, the note said that on the island one would find a gallows, an oak tree, and a
pine tree. To locate the treasure one would begin at the gallows, walk to the pine tree, turn
right 90 degrees and walk the same number of paces away from the pine tree. A spike was to
be driven at that point. Then return to the gallows, walk to the oak tree and turn left 90 degrees
and walk the same number of paces away from the oak tree. Drive a second spike in the ground.
The midpoint of a string drawn between the two spikes would locate the treasure.
Suppose the two trees are 100 feet apart. The location of the gallows is unknown, but it turns
out that any location of the gallows leads to the same spot for the treasure. Make a scale
drawing, and place the gallows at three different locations to determine the location of the
treasure. Where is the treasure buried relative to the two trees?
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8.
Explain why the treasure is always at the same location regardless of where the gallows is
located. Hint: Draw perpendiculars from the points used in locating the treasure to the line
between the two trees and consider congruent right triangles in the diagram.
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