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POW Solution, Jan. 31 - Feb. 4
Posted:
Feb 7, 1994 5:36 PM
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**********************************************************************
Problem of the Week for January 31 to February 4
Two engineers are scouting out an area where they plan to build a
bridge. They are standing on the edge of the river, directly across
from where the other end of the bridge will be. Their problem is
this: They don't know how far it is across the river. They puzzle
about this for a minute, and then Jill says to Jimmy, "I know how
to find out how far it is."
D Jill and Jimmy walk along the edge of the
----------------*-------- river to point B. Jimmy stays there. Then
Jill walks to C, making sure that the distance
river from B to C is equal to the distance from A
to B. Then she turns 90 degrees, and walks
----*-----*-----*--------- away from the river until Jimmy, who is
C| B A standing at B, is directly in line with D,
| the point where they want the other end of
| bridge. This is point E. Jill claims that
| the distance from C to E is the same as the
* distance from A to D. Is she right? Why
E or why not?
**********************************************************************
This proved to be a popular, or at least timely, puzzle - about 70
responses were submitted! Correct solutions were received from
(names are listed in the order in which the solutions were received):
Tammy Manski Grade 10, Shaler Area High School
Susan Quan Grade 7, Masterman School, Philadelphia
Nick Szmyd Grade 10, Shaler Area High School
Bipin Mujumdar Grade 10, Shaler Highschool
Ian Ross Grade 10, Shaler High School
Todd Gatnarek Grade 10, Shaler High School
Drew Ludwig 10th grade, Shaler High School
Kate Rusbasan Grade 10, Shaler High School
Julie Jadlowiec Grade 10, Shaler Area High School
Eric Carlson Grade 10, Shaler Area High School
Liz Boal Grade 10, Shaler Area High School
Jennifer Beranek Grade 10, Shaler Area High School
Hilary Aleksa, grade 9 and Anna Mata, grade 9, Fairfield High School
Others from Fairfield High School, Fairfield, Connecticut:
Grade 9 Grade 10
Amanda Adams Elissa Colter
Lindsey Becker Emily Leamy
Amy Decrescenzo Matt Lucas
Scott Dwyer Morgan May
Brendan Hogan Natalie Painchaud
Molly MacDonald Ryan Phelan
Kathy Medlin Tim Schnurr
Rebecca Naughton Nghiem Vu
Agata Raszczyk-Lawska
Phil Rossi
Katy Ruff
Allison Sullivan
Heather Wilcox
January Wilson
Lauren De Julio 10th Grade, Shaler Area High School
Ernie Leonetti 10th grade, Shaler Area High School
Ted Stevens 9th Grade, Steel Valley High School
Bob Gallagher & Ryan Ferchak, grade 9, Steel Valley High School
Mike Gibson Grade 10, Edgerton High School, Wisconsin
Kelly Donahue and Patti Dorn, Grade 11, Edgerton High School, WI
Andy Bilhorn Grade 10, Edgerton High School, Wisconsin
Jeremy Goede Grade 10, Edgerton High School, Wisconsin
Laureanna Raymond Grade 10, Edgerton High School, Wisconsin
Daniel Chan grade 10 student at Burnaby South SS, Burnaby BC Canada.
Others from Burnaby South SS, Burnaby BC Canada:
Annie Chan Gr 9 Richard Ho Gr 8
Cindy Ho Gr 8 Joe Bonifacio Gr 8
Melodie Chan Gr 8 Emily Wu Gr 8
Sonia Mezo Gr 8 Yi-Teng Huang Gr 8
Amelia Chia Gr 8 Les Shay Gr 8
Jennifer Strong Grade 10, Shaler Area High School
Some, but not all of the solutions, are included below. Those included
were the strongest in terms of wording and accuracy. Those which were
not included were lacking small details, or ideas were misapplied. I
have tried to respond to everyone whose proof was omitted from the list
below, and have hopefully given them some pointers which will help them
to make clearer arguments in the future.
Unfortunately, with this number of submissions, it is cumbersome to
include all the responses we received.
**********************************************************************
From: tammy manski <manski@one.sasd.k12.pa.us>
Jill is correct if the bridge is perpendicular to the river, forming
a right angle at point A and if the river is straight.
The distance between C and B is equal to the distance between A and B
because of the given information. Angle DAB is congruent to angle
BCE also because of the given information and the assumption that the
bridge forms a right angle with the edge of the river. Because D, B,
and E & C, B, and A form lines, angles CBE and DBA are vertical
angles (Two angles are vertical angles if their sides form two pairs
of opposite rays.) These two angles are congruent because of the
Vertical Angle Theorem that states that vertical angles are
congruent. Points B, C. and E form a triangle and points A, B, and D
also form a triangle. Triangles ECB and DAB are congruent because two
angles and the included side of those angles are congruent (All
triangles with an ASA correspondence are congruent.) This means that
CE=DA because corresponding parts of congruent triangles are equal
and Jill is correct.
Tammy Manski
Grade 10
Shaler Area High School
**********************************************************************
From: bipin mujumdar <mujumdar@one.sasd.k12.pa.us>
At the start of the problem I realized that I could prove she was
right by using an ASA correspondence. To do come at this conclusion
I did the following:
First I knew that the 2 right angles were congruent because all
right angles are congruent..
The distances between BC and BA were equal because that was given.
THe third pair of angles were congruent because they were vertical
angles and vertical angles are congruent.
After proving those pairs congruent I proved the two triangles
congruent by Angle Side Angle.. Finally I knew that DA equaled CE
because corresponding parts of congruent triangles are congruent. My
conclusion was that Jill was that her method for finding the distance
across the river worked.
Bipin Mujumdar, Shaler Highschool Grade 10
**********************************************************************
From: m2 <m2@one.sasd.k12.pa.us>
If it is assumed that the river is straight and that CA is
perpendicular to DA, then angle DAC will be a right angle. It is
given that CB will equal BA and angle CBE is congruent with angle ABD
because they are verticle angles. Since the two right angles are
congruent, the triangles are congruent by ASA. The length DA across
the river is equal to CE by CPCTC. Jill's theory is correct.
*Note that if DA is not perpendicular to CA Jill's theory
will not be correct because the two triangles will not be congruent.
**Also note that if the river is not straight Jill's theory
will also be false because it was not given that ABC were collinear.
If these points are not collinear her theory will not work.
Todd Gatnarek, Shaler High School, Grade 10
**********************************************************************
From: m1 <m1@one.sasd.k12.pa.us>
Drew Ludwig 10th grade Shaler High School
Jill and Jimmy's Idea would work, as long as the river is straight,
and they make sure that the turn at right angles at points A and C
For a curved river, Jill would need to wade into the water. The
reason their solution would work for a straight river is that the
triangles that Jimmy and Jill form is congruent by angle side angle.
Angle A is congruent to angle C, because they are both right angles.
Angle DBA is congruent to angle EBC because they are vertical angles.
CB is congruent to BA because Jill walked the same distance to form
both.
P.S. Jimmy sure is mean to make Jill do all the walking. What's
wrong with him?
**********************************************************************
From: m5 <m5@one.sasd.k12.pa.us>
Hello! My name is Liz Boal and I am a sophomore at Shaler Area
High School. Here is my solution to the problem of the week.
First we must assume that the river in the problem is a straight
line. We know that Jill made sure that segment BC is congruent to
segment AB. We are also told that angle A and angle c are right
angles. Angle A and angle C are congruent because any two right
angles are congruent. Next we have to introduce segment ED. We can
now say that angle CBE is congruent to angle DBA using the vertical
angle teorem. By using the angle-side-angle theorem we can prove
that triangle CBE is congruent to triangle ABD. Segment CE is
congruent to segment AD because congruent parts of congruent
triangles are congruent. Jill is right and her claim is true.
**********************************************************************
From: m4 <m4@one.sasd.k12.pa.us>
Hi, I'm Jennifer Beranek. I am in tenth grade at Shaler Area High
School. This is my solution to the problem of the week.
First, we must assume that angle DAB forms a right angle, and that
the river is straight. From the given information, CB is congruent
to AB. Angle ECB is congruent to angle DAB by definition of right
angles. By the line postulate, we introduce auxillary line ED.
Angle EBC is congruent to angle DBA by the vertical angle theorem.
From this information, trangle CBE is congruent to triangle ABD by
angle-side-angle theorem. Now, segment CE is congruent to segment AD
because corresponding parts of congruent triangles are congruent.
Jill is right in her claim that the distance from C to E is the same
as the distance from A to D.
**********************************************************************
From: PDALEY@fair1.fairfield.edu
The distances CE and AD are equal. By drawing a line through B from E to D,
two congruent triangles are formed. There are congruent vertical angles,
angle CBE and angle ABD. AB is congruent to CB because Jill went the
same distance from Jim as Jim went from the location of the bridge. DA is
perpendicular to BA because the shortest distance between two lines is the
perpendicular segment connecting them. Therefore, angle DAB is congruent to
angle ECB. Triangle DAB is congruent to triangle ECB by Side-Angle-Side.
AD is congruent to CE because corresponding parts of congruent triangles
are congruent.
Hilary Aleksa, grade 9 and Anna Mata, grade 9
Fairfield HS, Fairfield, CT
**********************************************************************
From: m1 <m1@one.sasd.k12.pa.us>
Lauren De Julio, 10th Grade, Shaler Area High School
Using the picture given:
Prove: DA=EC
1. AB=CB, <BCE Right Angle 1. Given
2. <DBA=<EBC 2. Vertical Angle Theorem
Assuming DA is Perpendicular to BA
3. <DAB=<ECB 3. Def. of Perpendicular
4. Triangle BCE=Triangle BAC 4. Angle Side Angle
Jill would be correct in saying that DA=EC by Corresponding Parts
of Congruent Triangles are Congruent
**********************************************************************
From: akuemmel@students.wisc.edu (Andrew Kuemmel)
Solution by Mike Gibson, Grade 10, Edgerton High School, Wisconsin
Yes, she is right.
Given: angle (BCE) = angle (BAD) = 90 degrees
BA = DC
segment (ED) contains B
Prove: DA = EC
1. m. angle (BCE) = m. angle (BAD) and BA=BC 1. Given
2. m. angle (EBC) = m. angle (DBA) 2. Vertical Angle Theorem
3. triangle (EBD) is congruent to triangle(DBA) 3. Angle Side Angle
4. EC = DA 4. corresponding parts
of congruent figures
**********************************************************************
From: akuemmel@students.wisc.edu (Andrew Kuemmel)
Solution by Andy Bilhorn, Grade 10, Edgerton High School, Wisconsin
The distances are equal. Both angle (DAB) and angle (BCE) are right
angles, and B is the midpoint of segment (AC). That would prove that
segments (AB) and (BC) are congruent. Angle (DBA) and angle (EBC) are
congruent because of the Vertical Angle Theorem. That makes triangle (DBA)
and triangle (EBC) congruent, because of the Angle Side Angle Theorem.
Both segments (CE) and (AE) are congruent because of the Corresponding
parts of congruent figures theorem.
**********************************************************************
From: akuemmel@students.wisc.edu (Andrew Kuemmel)
Solution by Jeremy Goede, Grade 10, Edgerton High School, Wisconsin
Jill is right because the two triangles (DAB) and (ECB) are congruent,
since AB = BC, angle (A) and angle (C) are right angles, and angle 9EBC)
is congruent to (DBA) because they are vertical angles. Therefore triangle
(DAB) is congruent to triangle (ECB) because of the Angle Side Angle
congruence theorem. Then segment (EC) is congruent to (AD) because of the
corresponding parts of congruent figures theorem.
**********************************************************************
From: akuemmel@students.wisc.edu (Andrew Kuemmel)
Solution by Laureanna Raymond, Grade 10, Edgerton High School, Wisconsin
To prove that the distance from C to E is equal to the distance from A to
D, I found you would have to prove the two triangles congruent. You cold
then use the corresponding parts of congruent figures theorem to solve the
problem.
1. angle (DBA) congruent to angle (EBC) 1. Vertical Angle Theorem
2. CB = BA 2. given
3. m. angle (C) = m. angle (A) = 90 3. given
4. triangle (CBE) congruent to triangle (ABD) 4. Angle Side Angle
5. CE = AD 5. cpcf
**********************************************************************
From: "Jim Swift" <jswift@cln.etc.bc.ca>
The following solution was provided by Daniel Chan, a grade 10 student at
Burnaby South SS, Burnaby BC Canada.
In the triangles CEB and ADB
/_ EBC = /_ DBA (given = 90 deg)
CB = BA (given)
/_ ECB = /_ DAB (vertically opposite angles)
Therefore the triangles are congruent (ASA)
Therefore CE = AB
and Jill is right.
**********************************************************************
From: strong jenn <strong@one.sasd.k12.pa.us>
My name is Jennifer Strong and I am in Mr. Detzel's Honors Geometry
class at Shaler High School. My solution for the problem of the week
is as follows:
According to the picture and the included text,
GIVEN: DAB is a right angle ,ECB is a right angle (90 deg.), AB=BC,
and D-B-E
PROVE: DA=EC
The vertical angle theorem shows, (assuming D-B-E to be true), that
angle ABD is congruent to angle CBE. Since all right angles are
congruent, DAB is congrent to ECB. Using the given information
AB=BC, one can then conclude that trianle DAB is congruent to
triangle ECB by Angle Side Angle. Since congruent parts of congruent
triangles are congruent, DA=CE.
*I used the lettering for the picture included with the problem
Therefore, Jill is correct that the distance from D-A is the same as
the distance from C-E. However, this would only work if D-B-E is
exactly correct.
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