Draw a good picture and carefully explain why you think your answer is right!

[This problem was sent to me by a teacher who was doing the PoW. He had shared it with some students in his class, and I thought I would give all of you a shot at it.]

Oh boy, two toughies in a row - I expected to catch a lot of you this week, but not this many! 26 people got this one right, and 126 got it wrong - of those 126, 96 gave the wrong answer that I thought people might give me.

The reason I knew that a lot of you would say FD=5 is because that's what I did when I first looked at the problem. I looked, said "5," and put it down. But then I turned back to it and said, "Hey, wait!" and right then I knew I had to use it for the PoW, and see how many of you would keep going after you found one answer.

A lot of you thought (and some of you said) that this problem was awfully easy, and that's because you stopped too soon! Nobody (least of all me) said that "the answer" meant a single number. There are going to be a lot of problems you come up against where it is easy to come up with an answer, but you have to be sure that you have ALL the answers. I didn't ask for a single answer, but for "the answer," and that means tell me everything you can. For example, if I asked you, "What number squared will give me 4?" everyone will say 2. And they'll all be right, but -2 isn't too hard to come up with either, and now you've found the whole answer. Geometry often works the same way - there will be a set of numbers that will work, and you need to find a way to describe that set of numbers.

Of course, finding other ways to split the parallelogram in half might not have been obvious to everyone. You need to play around with a picture. The way that Jennifer Champine of Southeast Raleigh High School described her answer is pretty good - you can get parallelograms, triangles, trapezoids, and rectangles when you split the parallelogram. That really helps you see the different ways you could divide it equally. You can read Jennifer's solution below.

Laura Roos of Shaler Area High School gave a solution that really explained how she went beyond the answer of 5. She made E the midpoint of AB, and the parallelogram a rectangle. So that gives you 5. Then she said what if ABCD wasn't a rectangle? FD is still 5. But NOW, what if E isn't the midpoint? Not many of you asked that question. Laura's solution is included for you to read.

In talking about how to split the parallelogram in half, Kevin Geier of Martin County High School simply explained that FD could be anything less than 10 as long as EB was the same thing. Aron Sturko of Pullman High School got a little more specific, saying that EF had to go through the intersection of the diagonals. Kelly Underkofler of Highland Park Senior High School included some very good pictures to illustrate these different ideas. But that's it - that's all the restrictions there are. Their solutions are all included below.

One thing you have to be careful about with a problem like this is that when you restate the problem and start to think about it, that you make sure you really stated the problem AS GIVEN. A lot of folks said things like, "...since EF bisects AB..." - wrong! It bisects the AREA of the parallelogram, but nobody said anything about bisecting AB. And nobody said that EF was parallel to AC and BD, either, so don't write it in your "givens". Another smaller point to make is that "between" (as in "E is between A and B") does not mean "midpoint." It simply means "between." If I meant midpoint, I would have said midpoint.

More than a few people got the idea that it could be different things, but they decided that the limits were from 3 to 7. I still don't understand quite how they got this, and find it interesting that a number of people got this same wrong answer. Push that parallelogram over a little more, and don't make EF perpendicular to AB.

The perfect illustration for a problem like this is a movable diagram on this Web page. If you have a Java-capable browser (and if I can get all this technology to work), you'll see a movable picture below. You can move any of the red dots. (This is made possible with a Java version of The Geometer's Sketchpad from Key Curriculum Press [http://www.keypress.com/sketchpad/java_gsp/].)

Sorry, this page requires a Java-compatible web browser.

A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.

From:   Jennifer Champine
        
Grade:  9
School: Southeast Raleigh High School, Raleigh, North Carolina

I've been trying it out, and because EF does not have to be parallel to 
any sides, but it could split the parallelogram in half into two 
triangles, two trapezoids, or two parallelogram, really FD could be 
anywhere from 0-10.
Jennifer

From:   Kevin Geier
        
Grade:  10
School: Martin County High School, Stuart, Florida

if AB=10, then CD=10

and segments AD and BC are parallel because it is a parallelogram.

E can lie on any point on segment AB.  F can lie on any point exactly opposite 
of E.  For example, if E is 3 away from point A on AB, then F has to be 3 away 
from C on CD.  Another example would be if E is 4 away from A on AB, then F has 
to be 4 away from point C on CD.  There is really no wrong answer as long as 
segment EF splits parallelogram ABCD's area in half....

so FD= anywhere from 1 to 10.

From:   Laura Roos
        
Grade:  10
School: Shaler Area High School, Pittsburgh, Pennsylvania

Subject: Answer to Geometry Problem of the Week February 9-13, 1998

Laura Roos
Grade 10
Shaler Area High School
Pittsburgh, PA

Geometry Problem of the Week
Splitting a Parallelogram
February 9-13, 1998

	I drew my parallelogram as a rectangle, because by definition, a rectangle 
is a parallelogram with four right angles.   If AB=10, then to split the 
rectangle in half, if E was the midpoint of AB, then F would be the midpoint of 
CD, making FD=5.  I tried it with a parallelogram that was not a rectangle and 
the same thing happens if E is the midpoint of AB.  Just say that E isn't the 
midpoint of AB, can it still work?  So I tried and it did.  The only thing was 
that the measure of FD changed too.  I believe that FD is equal to any number 
greater than 0 but less than 10 because AB is 10.  If E is placed anywhere on 
line AB, and the slope of the line changes, then EF will always divide the 
parallelogram in half.  Each time the parallelogram will have different measures 
for BC and AD.

From:   Kelly Underkofler
        
Grade:  9
School: Highland Park Senior High School, St. Paul, Minnesota

Subject: Feb 13 POW

Kelly Underkofler, grade 9, Geometry IB
Highland Park Senior High School, (612) 293-8940
www.stpaul.k12.mn.us/hphs/highland.html
February 9-13 POW

From:   Aron Sturko
        
Grade:  
School: Pullman High School, Pullman, Washington

Subject: LATE SUBMITIONS FROM ARON STRUKO

Splitting a Parallelogram
Feb. 9-13,1998

Question:
	What is FD?

Given:
	AB=10
	EF=10
	EF Divides area to ABCD in half
	F lies on CD

Solution:
	AB=CD	Oppsite siides of Parallelogram are
congurent.
For EF to Divide area by half it must pass through the
point of intersection of bisecting diagonals.

AD or CB divide area of ABCD in half - by Definition of
biseting diagonals.

The location of point F can vary Between C and D.
Therefore 0  FD  10

The following students submitted correct solutions this week: