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POW Solution, Jan. 31  Feb. 4
Posted:
Feb 7, 1994 5:36 PM


********************************************************************** 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 RaszczykLawska 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 YiTeng 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 anglesideangle 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 anglesideangle 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 SideAngleSide. 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 DBE PROVE: DA=EC
The vertical angle theorem shows, (assuming DBE 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 DA is the same as the distance from CE. However, this would only work if DBE is exactly correct.



