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Geometry POW Solution, December 15-19

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Geometry Problem of the Week

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Jan 15, 1998, 3:00:00 AM1/15/98

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Problem of the Week, December 15-19

A couple of weeks ago we explored the areas of different pieces of a
tangram. We also looked at congruent triangles in the tangram. A number
of people said that one triangle was congruent to another triangle
"because their areas are equal." Here is my question:

If two triangles have equal area, does that mean that they are
congruent? If your answer is yes, explain why. If your answer is no,
provide some "counter examples", that is, give me some triangles that
have equal area but are not congruent.

**************************************************

This was an inspired problem! Inspired mostly by some of the solutions
that I received on the tangrams problem, but it's also just fun. I
really liked it, and I guess many of you found it manageable, because 156
of you got it right, and 37 got it wrong. Of those 37, 35 had the right
idea, but made what I considered an "inexcuseable error".

Let me explain what two of those errors were. First, if you got the
areas of your triangles wrong, no credit. This mostly happened when
folks forgot to divide by two. The other was when the solution contained
"impossible" triangles, meaning that the triangle couldn't possibly
exist. This included giving a hypotenuse for a right triangle that
wasn't right. There is an easy way around something like this - don't
give the numbers you don't need. It's enough to say that a right
triangle has a base of x and a height of y. The hypotenuse doesn't
matter at all. Same goes for other triangles - you don't have to give
the legs of an isosceles triangle, just give the height and base. That's
all that matters.

There were a number of ways to provide counter examples. One is to give
two right triangles that have different bases and heights, but equal
areas. Simple enough. Another is to give two triangles with the same
base and height, but with different shapes - such as a right triangle and
an isosceles triangle.

Alex Morgovsky of Akiba Hebrew Academy leads off our highlighted
solutions this week. For one, he was right and provided three
well-presented counter-examples. For another, he admitted that he
actually learned something! In fact, a number of you mentioned that you
learned something this week. That's super!

Lev Navarre of Odle Middle School gave two ways of doing it. One is by
using a parallel line to draw _many_ triangles with the same base and
height. The other was to draw lots of rectangles with the same areas,
and split them in half. This brings up the idea of "factors" - different
pairs of numbers that multiply together to make the same number. Once
you get that, you've got this problem nailed.

Dane Skilbred of North Pole High School drew two nice triangles with text
characters - a good skill to have! He also got the problem correct :-)

Alison Miller, who is homeschooled, also tackled the parallel line idea,
and provided a really nice picture of it. She also talks of other ways
to get the counter examples. Check it out below.

Now, if you have a decent web browser, you MUST go look at Tim Peterson's
solution on his web page. He's got the parallel line thing going, and IT
MOVES! It's very cool, so check it out.

One comment that I really liked was provided by Jen Ramirez of Germantown
Academy. Jen said, "However, if the statement, 'If two triangles are
equal in area, the triangles are congruent,' was used in a sometimes,
always or never problem (*hehe*), the answer would be sometimes." This
is an excellent example of that! They might be congruent, but they don't
have to be. That is the point of a counter example. If you can find
_one_ triangle that doesn't work, then it can't be "always".

***********************************************

The following students submitted correct solutions this week.
Highlighted solutions are included below. For a full list of solutions,
please check out
http://forum.swarthmore.edu/geopow/fullsolutions/19971219.fullsolution.html.

Alex Morgovsky, Grade 11, Akiba Hebrew Academy, Merion, Pennsylvania
Brian Gainor, Grade 8, Perkiomen Valley Middle School, Collegeville,
Pennsylvania
Jason Chiu, Grade 9, Laramie Junior High School, Laramie, Wyoming
Alex Doskey, Grade ,
Tracy Steed, Grade 12, Wilburton High School, Wilburton, Oklahoma
Tiffanie Lam, Grade 8, Sequoia Middle School, Pleasant Hill, California
Robert Witting, Grade 9, Granada High School, Livermore, California
Megan Nowak, Grade 8, Odle Middle School, Bellevue, Washington
Sarah Nowak, Grade 8, Odle Middle School, Bellevue, Washington
Nicole Sweet, Grade 10, Concordia Lutheran High School, Tomball, Texas
Adedayo Lesi, Grade 8, Cherokee Heights Middle School, Madison, Wisconsin
Adebanke Lesi, Grade 9, West High School, Madison, Wisconsin
Lev Navarre, Grade 6, Odle Middle School, Bellevue, Washington
Emily O'Brien, Grade 10, School Without Walls, Washington, DC
Gabriel Felix, Grade 9, South High School, Bakersfield, California
Desiree Benavides, Grade 9, South High School, Bakersfield, California
Perry Cottrell, Grade 9, Smoky Hill High School, Aurora, Colorado
Mike Redmond, Grade 9, South High School, Bakersfield, California
Chris Haenelt, Grade 9, South High School, Bakersfield, California
Brandon McDonel, Grade 9, South High School, Bakersfield, California
Stefanie Hill, Grade 9, South High School, Bakersfield, California
Jonathon Holland, Grade 9, South High School, Bakersfield, California
Brody Beecher, Grade 9, South High School, Bakersfield, California
Mike Young, Grade 9, South High School, Bakersfield, California
Jeff Wilson, Grade 9, South High School, Bakersfield, California
Matt, Grade 9, South High School, Bakersfield, California
Paul Scheideman, Grade 9, South High School, Bakersfield, California
Oanh Chi, Grade 9, South High School, Bakersfield, California
Sarah Pearson, Grade 9, South High School, Bakersfield, California
David Shaw, Grade 9, Smoky Hill High School, Aurora, Colorado
Milan Fillmore, Grade 9, Smoky Hill High School, Aurora, Colorado
Gordon Bockus Jr., Grade Freshman, Eastern Oklahoma State College,
Wilburton, Oklahoma
Ian Cochran, Grade 9, South High School, Bakersfield, California
Amy Neville, Grade 9, South High School, Bakersfield, California
Colin Shields, Grade 9, South High School, Bakersfield, California
Kali Rothrock, Grade 10, Delaware County Christian School, Newtown Square,
Pennsylvania
Brian, Grade secondary ed. student, Buffalo State College, Buffalo, New York
Brad Ross, Grade Freshman, Kennesaw State University, Kennesaw, Georgia
Howard Sun, Grade 6, Lynngate Junior Public School
Dane Skilbred, Grade 9, North Pole High School, North Pole, Alaska
Katie Anthony, Grade 9, Casady School, Oklahoma City, Oklahoma
Derek Howles and Matt Bruce and Ryan Ruch, Grade 8, Mont Pleasant Middle
School, Schenectady, New York
Lisa, Grade 10, Smoky Hill High School, Aurora, Colorado
Justin Pearson, Grade 9, Livermore High School, Livermore, California
Rick and Matt, Grade 10 & 9, Smoky Hill High School, Aurora, Colorado
Jennifer Nelson, Grade 9, Smoky Hill High School, Aurora, Colorado
Zach Rentz, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania
Jenny Lurie, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania
Jackie Evans, Grade 9, Smoky Hill High School, Aurora, Colorado
Erik Van De Vreugde, Grade 10, Livermore High School, Livermore, California
Jim Nguyen, Grade 9, Smoky Hill High School, Aurora, Colorado
Catherine Mangasi, Grade 12, Wilburton High School, Wilburton, Oklahoma
Robert Hencke, Grade 10, Francis Parker Charter School, Ayer, Massachusetts
The Math Mob, Grade 6, Ridge Mills Elementary School, Rome, New York
Andy Sebold, Grade 10, Concordia Lutheran High School, Tomball, Texas
Anna Carmack, Grade 10, Shelby County High School, Columbiana, Alabama
Derrick McGinnis, Grade 12, Shelby County High School, Columbiana, Alabama
Laura Harmacek, Grade 8, Challenge School, Denver, Colorado
Nathan Countryman, Grade 8, Challenge School, Denver, Colorado
Brandi Moore, Grade 8, Challenge School, Denver, Colorado
Darren Kerstien and John Yi, Grade 7 & 8, Challenge School, Denver, Colorado
Allison Bultemeier, Grade 10, Concordia Lutheran High School, Tomball, Texas
Rob Eagle, Grade 8, Challenge School, Denver, Colorado
Nicole Benroth, Grade 10, Willard High School, Willard, Ohio
Josh Taft, Grade , Smoky Hill High School, Aurora, Colorado
Roger Dieterich III, Grade 10, Smoky Hill High School, Aurora, Colorado
Kevin Yurkerwich, Grade , Okemo Mountain School, Ludlow, Vermont
Matt Simcox, Grade 9, East Mecklenburgh, Charlotte, North Carolina
Hunter Brooks, Grade 8, Camelot Academy, Durham, North Carolina
John Marion, Grade 9, Livermore High School, Livermore, California
Kaitlin Primavera, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania
Noam Abrams, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania
Katie Madden, Grade 9, Mount Saint Joseph Academy, Flourtown, Pennsylvania
Andrew Mundschau, Grade 9, South High School, Bakersfield, California
Whitney A. and Lauren Z., Grade 9, Germantown Academy, Fort Washington,
Pennsylvania
Mandeep Singh, Grade 9, Interlake High School, Bellevue, Washington
Colleen Kelly, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania
Danny McKenna, Grade 10, William Penn Charter School, Philadelphia,
Pennsylvania
Lauren O'Garro-Moore and Julie Lewis, Grade 10, Mount Saint Joseph
Academy, Flourtown, Pennsylvania
Chris Lauber, Grade 9, Smoky Hill High School, Aurora, Colorado
Katie Hathaway, Grade 10, Concordia Lutheran High School, Tomball, Texas
Jon Amt, Grade 10, Concordia Lutheran High School, Tomball, Texas
Jane Milton and Sara Fitzsimmons, Grade 10, Mount Saint Joesph Academy,
Flourtown, Pennsylvania
Ben Fox, Grade 10, Concordia Lutheran High School, Tomball, Texas
Margaret MacKenzie, Grade 10, Concordia Lutheran High School, Tomball, Texas
Tyson Becker, Grade 10, Concordia Lutheran High School, Tomball, Texas
Le Tran, Grade 10, Smoky Hill High School, Aurora, Colorado
Matthew Hallien, Grade 10, Concordia Lutheran High School, Tomball, Texas
Christy Thornburg and Katy Crumpton, Grade 10, Shelby County High School,
Columbiana, Alabama
Phummarin Sritongsook, Grade 9, South High School, Bakersfield, California
Avrum Tilman, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania
Jacob Ornelas, Grade , Smoky Hill High School, Aurora, Colorado
Julia Le, Grade 11, Minnechaug Regional High School, Wilbraham, Massachusetts
Kamila Sikora, Grade 9, Smoky Hill High School, Aurora, Colorado
Laura Meyer, Grade 10, Concordia Lutheran High School, Tomball, Texas
Alex Chen, Grade 7, Odle Middle School, Bellevue, Washington
David Grant, Grade , Livermore High School, Livermore, California
Scott , Grade 9, East Mecklenburg High School, Charlotte, North Carolina
Abby Jones, Grade 10, Smoky Hill High School, Aurora, Colorado
Megan Caulder, Grade 9, Granada High Scool, Livermore, California
James Lin, Grade 9, Southern California
Steven Connor, Grade 10, Pleasant Valley High School, Brodheadsvlle,
Pennsylvania
Delia Ryan, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania
Arlene Taylor, Grade 8, Lyneham, Canberra, Australia
Jenny Kaplan, Grade 7, Castilleja Middle School, Palo Alto, California
Thuy Nyuyen, Grade 11, Highland Park Senior High School, St. Paul, Minnesota
Jessica Pea, Grade 10, St. Louis Career Academy, St. Louis, Missouri
Shelli Delp, Grade Senior, Northeastern State University, Oklahoma
Keith Cusson, Grade 10, St. Johns School, Montreal, Quebec, Canada
Jeffrey DeVault, Grade 10, Delaware Country Christian School, Newtown
Square, Pennsylvani
Vibha Balu, Grade 9, Edison High School, Edison, New Jersey
Ellen Samuel, Grade 10, Oak Park and River Forest High School, Oak Park,
Illinois
Joan Dabu, Grade 9, Edison High School, Edison, New Jersey
Neil Seifried, Grade 10, Pullman High School, Pullman, Washington
Clayton Dillaway, Grade 8, Odle Middle School, Bellevue, Washington
Dave Espenshade, Grade , Washington
Geometry Class, Grade , Highland Park Senior High School, St. Paul, Minnesota
Chris Coelho, Grade , Cheshire High School, Cheshire, Connecticut
Aaron Howard, Grade 9, Highland Park Senior High School, St. Paul, Minnesota
Eric Collins, Grade 9, Highland Park Senior High School, St. Paul, Minnesota
Paul Kreiger, Grade 11, Highland Park Senior High School, St. Paul, Minnesota
Alison Falkenhagen, Grade 9, Highland Park Senior High School, St. Paul,
Minnesota
Scott Kling, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Jennifer Au, Grade 9, Germantown Academy, Fort Washington, Pennsylvania
Ron Rothrock and Eileen Rothrock and Kali Rothrock and Dana Rothrock, Grade ,
Bethany Kim, Grade , Redmond High School, Redmond, Oregon
Allen Hsu and Mike Sands, Grade , Nitschmann Middle School, Bethlethem,
Pennsylvania
Andrea Dexter-Rice, Grade , Nitschmann Middle School, Bethlehem, Pennsylvania
Jen Ramirez, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Kelly Washington, Grade 9, Germantown Academy, Fort Washington, Pennsylvania
Adam Fackler, Grade 11, Cheshire High School, Cheshire, Connecticut
Casey Sutherland, Grade , Redmond High School, Redmond, Oregon
Long Le, Grade , Granada High School, Livermore, California
Joanna Mack and Joanna Frankel, Grade 9, Germantown Academy, Fort
Washington, Pennsylvania
Darrin Koski, Grade , Redmond High School, Redmond, Oregon
Emily Buzicky, Grade 9, Highland Park Senior High School, St. Paul, Minnesota
Rashida Fisher, Grade 9, Highland Park Senior High School, St. Paul, Minnesota
Brandon Gilchrist, Grade 9, Highland Park Senior High School, St. Paul,
Minnesota
Ryan Holcomb, Grade , Redmond High School, Redmond, Oregon
Conor Ferguson, Grade , Redmond High School, Redmond, Oregon
Deidre Cohen, Grade , Redmond High School, Redmond, Oregon
Patty Eng, Grade , Cheshire High School, Cheshire, Connecticut
Alex Chernyavsky, Grade , Akiba Hebrew Academy, Merion, Pennsylvania
Joe Thomer, Grade , Germantown Academy, Fort Washington, Pennsylvania
Matthew Harrison, Grade , Germantown Academy, Fort Washington, Pennsylvania
JAR 82, Grade ,
Alison Miller, Grade 6, homeschooled, Niskayuna, New York
Tim Peterson, Grade , homeschooled, Rochester, New York
Thomas Kuo, Grade 10, Burroughs High School, Ridgecrest, California
Denny Chao, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Jennifer Liang, Grade 8, Odle Middle School, Bellevue, Washington
Jessica Barclay-Strobel, Grade 10, Oak Park and River Forest High School,
Oak Park, Illinois
Tracy Kennedy, Grade , Cheshire High School, Cheshire, Connecticut
Chaim Bloom, Grade ,
Lauren Rossi and Anne Hines, Grade , Germantown Academy, Fort Washington,
Pennsylvania
Charlie Beigarten, Grade 10, Granada High School, Livermore, California

***********************************************

From: Alex Morgovsky
pats...@aol.com
Grade: 11
School: Akiba Hebrew Academy, Merion, Pennsylvania

Triangles that have equal areas are not always congruent to each other. They
can be, but they don't have to be.

Counter example 1:
scale triangle ABC. isoscles triangle DEF
BC, the base, is 5 cm. EF, the base, is 20 cm.
The height is 10 cm. The height is 2.5 cm.
Area = bh/2 = 25 cm^2. Area = bh/2 = 25 cm^2.

Counter example 2:
isosceles right triangle GHI. isosceles triangle JKL
HI, the base, is 3 cm. KL, the base, is 9 cm.
The height is also 3 cm. The height is 1 cm.
area = bh/2 = 4.5 cm^2. area = 4.5 cm.

Counter example 3:
scale triangle MNO isoscleses triangle PQR
NO, the base, is 9 cm. QR, the base, is 100 cm.
The height is 1 cm. The height is .09 cm.
Area = bh/2 = 4.5 cm^2. Area = bh/2 = 4.5 cm^2.

The are extremely many counter-examples.

An interesting fact that I picked up from the last 2 weeks is that 2
figures can
have the same area, and can different numbers of sides.

***********************************************

From: Tiffanie Lam
quan...@ucop.edu
Grade: 8
School: Sequoia Middle School, Pleasant Hill, California

The best way to demonstrate a counter-example of non-congruent
triangles with the same area is to draw two parallel lines
C D
********************************
* * * *
* * *
* * * *
** * *
********************************
A B

Pick any two points on the bottom line and label them A and B.
These two points will form the base for the two non-congruent
triangles. Now Pick another two points from the top line and
call them C and D. Then Triangle ABC and Triangle ABD are not
congruent because their sides have different length, but they
do have the same area because they share the common base AB and
because the two lines are parallel, they also have altitude of
the same length.

***********************************************

From: Lev Navarre
levna...@hotmail.com
Grade: 6
School: Odle Middle School, Bellevue, Washington

I have 2 solutions.

1: Draw a triangle. Label it ABC. Draw a line parallel to BC
that passes through A, and call it AD. Now you can move point
A anywhere you want along the line AD, and the area of ABC will
stay the same. This happens because the equation for the area
of a triangle is area = 1/2 of the base times the height. The
base will obviously always stay the same, and therefore the 1/2
base in the equation will. It doesn't matter where A is as long
as it is on AD, because the height will always stay the same
since AD is parallel to BC.

2: Draw several rectangles whose area are 100 square units, such
as 1 by 100, 2 by 50, 4 by 25, 5 by 20, and so on. Then take
each rectangle and cut it along the diagonal, so that you have
many triangles, and ALL of them will be different and ALL of them
will have the same area, since ALL of the rectangles were 100
square units each.

***********************************************

From: Dane Skilbred
01...@student.northstar.k12.ak.us
Grade: 9
School: North Pole High School, North Pole, Alaska

No. The triangles don't necessarily mean they have to be congruent. For
example:

|\ /|\
4 | \ 5 / | \
| \ / |4 \
|___\ /___|___\
3 ~~~~~~~
3

The areas of these two triangles are the same, but they are not congruent. For
two triangles to be congruent they must have corresponding sides and angles.

***********************************************

From: Geometry Class
dpea...@mail.stpaul.k12.mn.us
Grade:
School: Highland Park Senior High School, St. Paul, Minnesota

Subject: Dec 19 POW

Summary of class discussion around Dec 19 POW
Highland Park Senior High School, (612)293-8940
www.stpaul.k12.mn.us/hphs/highland.html
October 6 - 10 Problem of the Week

My geometry class has just completed a unit of study about triangle congruence.
We have also learned to compute the areas of triangles in several different
ways. These two ideas came together in this week's POW. Here is a
part of our
class discussion.

<h4>If two triangles are congruent, then they are equal in
area.</h4>

This statement is true, and we can "prove" it by using
Heron's area formula that
we studied last week.

The "proof" for the above statement would go like this: If two
triangles are
congruent, then by definition, all six pairs of corresponding parts are
congruent and, in particular, all three pairs of corresponding sides are
congruent. If two triangles have the same side lengths, then Heron's
formula
would find their areas to be equal.

<h4>If two triangles are equal in area, then they are
congruent.</h4>

This statement is false., and we can see that by refering the midpoint theorem
we studied last week.

Triangle AMB and triangle CMB serve as counterexamples to show that the
statement above is false. These triangles have the same areas (same length
base,
same altitude), but they are not congruent (one is acute, one is obtuse, so all
of their corresponding parts cannot be congruent).

<h4>A footnote of logic</h4>

It is sometimes true that a statement and its converse are both true. It is as
often true that a statement is true but its converse is not. The two statements
we are discussing are converses.
If two triangles are congruent, then they are equal in area.
If two triangles are equal in area, then they are congruent.

It is easy to believe that if two triangles are congruent, then they must be
equal in area. This does not imply, however, that two triangles equal in area
must also be congruent. A statement does not imply its converse!!!

***********************************************

From: Alison Miller
mary_o...@classic.msn.com
Grade: 6
School: homeschooled, Niskayuna, New York

Subject: Dec. 13 Geometry POW

Dear Annie,

My answer is no. The formula for the area of a triangle is 1/2*(length of
base)*(length of altitude). Therefore, if two triangles share the same base,
but have different altitudes that have the same length, they have the same
area. My sketch illustrates that. You could also have two triangles with
different bases with the same length, and different altitudes with the same
length. Or, you could have triangles with the same altitude, but different
bases with the same length. Finally, you could have two triangles whose bases
are different lengths, whose altitudes are different lengths, BUT the product
of the lengths of the base and of the altitude of the first triangle is equal
to the product of the lengths of the base and of the altitude of the second
triangle. Those are all the possibilities.

Happy Holidays!

Alison Miller, Grade 6
Homeschooled, Niskayuna, NY

***********************************************

From: Tim Peterson
sfs...@prodigy.com
Grade:
School: homeschooled, Rochester, New York

Subject: POW 12/19

My answer is at
<a href="http://pages.prodigy.com/SFSS74C/gp121997.htm">
http://pages.prodigy.com/SFSS74C/gp121997.htm</a>

Two triangles have the same area if they have the same base and height. None of
the triangles with the same base and height are congruent because the
angles are different. Here are some triangles that have the same area but are
not congruent:

[animated picture of triangle with static base and height, but top moves
along a
line parallel to the base.]

And here are some triangles that have the same area and are congruent:

Tim Peterson

***********************************************

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