Geometry Forum - Problem of the Week
Solutions - Jill and Jimmy - Distance Across River, Jan. 31-Feb. 4, 1994
Tammy Manski
____________________
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
Bipin Mujumdar
____________________
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 High School, Grade 10
Todd Gatnarek
____________________
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
Drew Ludwig
____________________
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?
Liz Boal
____________________
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.
Jennifer Beranek
____________________
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.
Hilary Aleksa and Anna Mata
____________________
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
Lauren De Julio
____________________
Using the picture given: Prove: DA=EC 1. AB=CB, angle BCE Right Angle 1. Given 2. angle DBA= angle EBC 2. Vertical Angle Theorem Assuming DA is Perpendicular to BA 3. angle DAB= angle 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
Mike Gibson
____________________
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) 1. Given
and BA=BC
2. m. angle (EBC) = m. angle (DBA) 2. Vertical Angle Theorem
3. triangle (EBD) is congruent to 3. Angle Side Angle
triangle(DBA)
4. EC = DA 4. corresponding parts of
congruent figures
Andy Bilhorn
____________________
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.
Jeremy Goede
____________________
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.
Laureanna Raymond
____________________
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
Daniel Chan
____________________
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.
Jennifer Strong
____________________
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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