This problem is due the Friday AFTER Thanksgiving - you've got a lot of
time, so do a careful job.
A _________________________ D
|* *|
| * * |
| * F * | This is a picture of a tangram, which is a
| * * | puzzle that consists of a square cut into
| * * | | seven pieces. Which of the triangles are
| * E * | | congruent to each other?
| * | *|G
| * * | * |
| H * * | * |
| * * | If AB = 1, what is the area of each of the
| * * * I | seven pieces?
| * * * |
B|*___________*____________|C
J
Explain how you figured out the areas - there are several ways to do it,
and I want to know how YOU did it! (See if you can do it without actually
figuring out the edgelengths of anything.) No explanation, no credit.
Annie says: Solutions
This problem is just the start of a string of problems that will stem from the tangram - we'll get a lot of mileage from this ancient puzzle, because some interesting ideas come out of these answers.
116 people got this one right, and 61 got it wrong. More than a few of those who got it wrong failed to say which of the triangles are congruent. You must read the problem carefully and make sure you answer all the parts! Sure, it's pretty easy to see which ones they are, but that doesn't mean you don't have to answer it.
One idea you can be sure of here is that all the areas you find will add up to one, and that is an excellent way to check and see if you're right. That's exactly what Aaron Mertz of Plum Grove Junior High School did, and as the first of many to do so, he gets his solution highlighted.
It's a good idea to make your answer easy to read. A good way to do this, especially with a problem that has many parts, is to present your answers in the beginning of the problem, then give the explanations. This lets the reader know where they are going. You can read Megan Ross' answer below for an example. Megan goes to East Mecklenburgh High School.
There are a number of ways to figure this problem out. One would be to figure out all of the edgelengths and then find the areas of each region from there. That's okay, but it's an awful lot of work. A number of people chose to avoid all the work and divided the whole square up into little triangles, then counted how many little triangles it took to make up each piece. For an example, read the solution of Melissa Branfman of Georgetown Day School.
I didn't elaborate much in this problem, and that's because "tangram" has a specific meaning. But it's still a good idea to state what assumptions you made. Alex Chernyavsky of Akiba Hebrew Academy started his solution off by saying, "...we can assume that in a tangram all is as it seems...", which is exactly right. You can read the rest of Alex's solution below.
When you are naming your congruent triangles, be careful of the order in which you put the letters. Order _does_ matter, after all. And don't round your figures too much, especially if you are going to use the edgelengths to find the areas. If you've got sqrt2, leave it like that. Don't use 1.4, or your areas won't work out to add up to one.
Now, one more idea. Just because two triangles have equal areas, that doesn't mean that they're congruent. In fact, that's interesting enough that I might make it into a POW later this year! Certainly the converse is true - if they're congruent, they have equal area, but not the statement itself.
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: Aaron Mertz Grade: 8 School: Plum Grove Junior High School, Rolling Meadows, Illinois The two large triangles in the upper left are obviosly 1/2 of the entire square, so the area of each of these is 1/4 units. It takes two of the medium size triangle in the lower left to make one of the large triangles. 1/2 x 1/4 = 1/8 units, the area of the medium triangle. It takes two of the smaller triangles to make one of the medium triangle. 1/2 x 1/8 = 1/16, the area of the smaller triangles. It takes two smaller triangles to make one square shown in the figure, so 2 x 1/16 = 1/8 units, the area of the square. It takes two smaller triangles to make the parallelagram. 2 x 1/16 = 1/8 units, the area of the parallelagram. Now, let's check our answer: 1/4 (AEB) + 1/4 (AED) + 1/8 (HJIE) + 1/8 (DGIF) + 1/8 (CGJ) + 1/16 (EIF) + 1/16 (BHJ) = 1 x 1 The congruent triangles are: ABE and ADE; BHJ and EIF Thanks!
From: Megan Ross Grade: 9 School: East Mecklenburg High School, Charlotte, North Carolina The triangles that are congruent to each other are triangle ABE and triangle ADE. Another pair of congruent triangles are triangle BHJ and triangle IEF. The area of each of the seven pieces are: Triangle ABE=1/4 Triangle ADE= 1/4 parallelogram FDGI =1/8 Triangle FEI= 1/16 Triangle BHJ =1/16 Triangle JCG= 1/8 Square HJIE =1/8 We got this by spliting the square into 4 congruent triangles. Triangles ABE and ADE are both equal to 1/4 of the entire square. The square HJIE is equal to 1/8 of the square. Triangle BJH is half the size of the box, so it is 1/16. If you split Triangle JCG in half to make 2 congruent triangles, they would each be 1/16, so add both of those together and you get 1/8. Triangle IEF is half of triangle JCG so it's area is 1/16. The area of parallelogram FIGD is 1/4 because you can create 2 congruent squares in it. Those squares are both congruent to triangle IEF whose area is 1/16. So, the area of the parallelogram is twice the area of triangle IEF, so it would be 2/16 or 1/8.
From: Becky Marsh Grade: 9 School: Kealakehe High School, Kailua Kona, Hawaii The area of triangle AED is .25 The area of triangle AEB is .25 The area of triangle JGC is .125 The area of FIGD is .125 The area of triangle EFI is .0625 The area of HEIJ is .125 The area of BHJ is .0625 Triangles AEB and AED are congruent Triangles BHJ and EFI are congruent I divided the square into 32 equal pieces by drawing lines verticaly, horizontaly, and diagonaly.When finished I had a square made up of 32 congruent triangles, and each shape inside the square was made up of two or more of these small triangles. Since I knew that these 32 triangles made up the square ABCD, I knew each triangle had an area of 1\32, or .03125. To find the area of each particular shape inside the triangle, I just found the number of these small triangles that made it up and multiplied this number by .03125.
From: Melissa Branfman Grade: 8 School: Georgetown Day School, Washington, DC Subject: problem of the week Geometry Problem of the week Melissa Branfman (Moodring5) Paul Nass Georgetown Day School Washington DC Eighth Grade Math I'm not sure if I am about to answer this weeks problem correctly, but I'll give it my all. First of all there are three sets of congruent triangles in this tangram. The first pair is the largest: triangle ABD and triangle CDB. Next, triangle AEB is congruent to triangle ACD. Thirdly triangle BHJ is congruent to triangle IEF. We know that the first pair of congruent triangle are congruent because they are formed by a square cut by a triangle. Therefore SSS is congruent to SSS (or any of the reasons for two triangle to be congruent will work). The other two triangle pairs are congruent because with tangrams there are two sets of congruent triangles: A large set and a smaller set. Now to find the area of each shape we first must look at the smallest (or one of the smallest) triangle. We can see how many times that it fits into each shape. The smallest trianlge will fit into triangle ABE four times. It will also fit into triangle AED four times. The triangle will obviously fit into triangles BHJ and EIF once each. The smallest triangle will fit into the square twice (square HJEI). It will fit into triangle JGC twice also. As well it will fit into polygon FDGI twice. All together the triangle will fit into all of the shapes to a sum of sixteen times. Therefore each small triangle is one sixteenth (if the side is one, but if you don't use number than it is one sixteenth of the whole. This will be true for all of the following shapes.). Triangle ABE and AED are both one fourth. Square HEJI and polygon FDGI and triangle JGC are all one eighth of the square. So that is the area of each shape.
From: Alex Chernyavsky Grade: School: Akiba Hebrew Academy, Merion, Pennsylvania Subject: Tangrams,congruent triangles,and area Since we can assume that in a tangram everything is as it seems,then triangle ABE is congruent to triangle ADE,and triangle BHJ is congruent to triangle FEI. Since the length of AB=1,and the length of the diagonal=the square root of 2,then area of ABE=1/4,area ADE=1/4,area of JGC=1/8;area of HEIJ=1/8;area of BHJ=1/16;area of FEI=1/16,and area of FDGI=1/8(You don't have to figure out the edgelengths to get the area because since we know that E,H,J,I,G,and F are all midpoints,we just take the given AB=1,get the diagonal-square root of 2,and then figure it all out by the formulas for area).
The following students submitted correct solutions this week:
Gabe Ho, Grade 10, North Quincy High School, Quincy, Massachusetts
Doug Brillinger, Grade 6, Queen Elizabeth Public School, Oshawa, Ontario, Canada
Jackie Evans, Grade 9, Smoky Hill High School, Aurora, Colorado
Alex Morgovsky, Grade 11, Akiba Hebrew Academy, Merion, Pennsylvania
Aaron Mertz, Grade 8, Plum Grove Junior High School, Rolling Meadows, Illinois
Matt Hazel, Grade 8, St. Gregory School
Tiffanie Lam, Grade 8, Sequoia Middle School, Pleasant Hill, California
Megan Ross, Grade 9, East Mecklenburg High School, Charlotte, North Carolina
Dane Skilbred, Grade 9, North Pole High School, North Pole, Alaska
Wendy , Grade 8, St. Ann School, Quincy, Massachusetts
Ryan Browning, Grade 9, Archmere Academy, Wilmington, Delaware
Tan Xiaojia, Grade Secondary 2, Nanyang Girls' High School, Singapore
Holly Black, Grade 7, Odle Middle School, Bellevue, Washington
John Martin, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania
Becky Marsh, Grade 9, Kealakehe High School, Kailua Kona, Hawaii
Natko Bajic, Grade 7, Pojisan Primary School, Split, Croatia
Jie Yu, Grade 9, Langley High School
Julia Fischer, Grade 10, Granada High School, Livermore, California
Bassel Rifai, Grade 10, Green Valley High School, Henderson, Nevada
Chad George, Grade , Nuclear Power School, US Navy
Ellen Samuel, Grade 10, Oak Park and River Forest High School, Oak Park, Illinois
Jenny Kaplan, Grade 7, Castilleja Middle School, Palo Alto, California
Harold Roa Cadeliņa, Grade 10, Notre Dame of Cotabato, Philippines
Gordon Bockus Jr, Grade 9, Eastern Oklahoma State College, Wilburton, Oklahoma
Math Mob, Grade 6, Ridge Mills Rome, New York
Paul Cates, Grade 8, Albright Middle School, Houston, Texas
Jim nguyen, Grade 9, Smoky Hill High School, Aurora, Colorado
Joe, Grade 5, homeschooled, Canterbury, New Hampshire
Ben Spielberg, Grade 4M, Stoy Elementary School, Haddonfield, New Jersey
Richard Henry, Grade 12, North Pole High School, North Pole, Alaska
Andy Swenson, Grade 9, North Pole High School, North Pole, Alaska
Jeff Serrano, Grade 10, North Pole High School, North Pole, Alaska
Jennifer Harmon, Grade 9, North Pole High School, North Pole, Alaska
Michael Smith, Grade 10, North Pole High School, North Pole, Alaska
E. Levinson and C. Greenberg, Grade 9, Germantown Academy, Fort Washington, Pennsylvania
Adam Fackler and Patty Eng and Katie Rock and Justin Coleman, Grade 11, Cheshire High School, Cheshire, Connecticut
Alex Doskey, Grade ,
Allen Hsu and Mike Sands, Grade , Nitschmann Middle School, Bethlehem, Pennsylvania
Doug Yoder, Grade 12, Highland Park Senior High School, St. Paul, Minnesota
Susan Wolff, Grade , Holton-Arms School, Bethesda, Maryland
Teneal Dollar, Grade 10, Shelby County High School, Columbiana, Alabama
Peter Geraldino, Grade , Germantown Academy, Fort Washington, Pennsylvania
Kimberly Woodall, Grade 10, Shelby County High School, Columbiana, Alabama
, Grade , Georgetown Day School, Washington, DC
David Zax, Grade 8, Georgetown Day School, Washington, DC
Emma Lindsay, Grade 8, Georgetown Day School, Washington, DC
Lucy Wimpenny, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Li-Yun Wang, Grade , Smoky Hill High School, Aurora, Colorado
Katy Crumpton, Grade 10, Shelby County High School, Columbiana, Alabama
Christy Thornburg, Grade 10, Shelby County High School, Columbiana, Alabama
Chester Chan, Grade 6, Odle Middle School, Bellevue, Washington
Colleen Kelly, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania
Math Mob, Grade 6, Ridge Mills, Rome, New York
Ashley Tierney, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania
Laura Harmacek, Grade 8, Challenge School, Denver, Colorado
Mark Van Arsdale, Grade 8, The Challenge School, Denver, Colorado
Darren Kerstien and Rob Eagle, Grade 7 and 8, The Challenge School, Denver, Colorado
Jessica Black and Brandi Moore and Kellyan Coors, Grade 8, The Challenge School, Denver, Colorado
Ben Olsen and Con Vigderman, Grade 8, The Challenge School, Denver, Colorado
Jeremy Smith, Grade 10, Cascade Christian Academy, Wenatchee, Washington
Melissa Branfman, Grade 8, Georgetown Day School, Washington, DC
Roger Dieterich, Grade 10, Smoky Hill High School, Aurora, Colorado
Joe Ciolek, Grade 8, Jefferson Junior High School
Bryan Baker, Grade 9, Smoky Hill High School, Aurora, Colorado
Neil Seifried, Grade 10, Pullman High School, Pullman, Washington
Leo Shimizu, Grade 6, Odle Middle School, Bellevue, Washington
David Fox, Grade 8, Gergetown Day School, Washington, DC
Le Tran, Grade 10, Smoky Hill High School, Aurora, Colorado
Jessica Arendal, Grade 8, Georgetown Day School, Washington, DC
Ariel Berenstein, Grade 8, Georgetown Day School, Washington, DC
Emily O'Brien, Grade 10, School Without Walls High School, Washington, DC
Jason Bryant, Grade 8, Georgetown Day School, Washington, DC
Nathan Countryman, Grade 8, The Challenge School, Denver, Colorado
Tracy Steed, Grade 12, Wilburton High School, Wilburton, Oklahoma
Lily Tian, Grade 8, The Challenge School, Denver, Colorado
rachel meyer, Grade 8, The Challenge School, Denver, Colorado
Chris Lauber, Grade 9, Smoky Hill High School, Aurora, Colorado
ctsui14, Grade 9, Smoky Hill High School, Aurora, Colorado
Katie Kelly and Tracy Kennedy and Neil Peterson and Nick Baxter, Grade , Cheshire High School, Cheshire, Connecticut
Matt Niederst, Grade , Shaler Area High School, Pittsburgh, Pennsylvania
Laura Roos, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania
Niki Weber, Grade , Shaler Area High School, Pittsburgh, Pennsylvania
Robin Thornburg and Anna Carmack, Grade 11 & 10, Shelby County High School, Columbiana, Alabama
Eric Lindberg, Grade 7, Odle Middle School, Bellevue, Washington
Melissa Branfman, Grade 8, Georgetown Day School, Washington, DC
Jen Erhart, Grade , Shaler Area High School, Pittsburgh, Pennsylvania
Tony Kambic, Grade , Shaler Area High School, Pittsburgh, Pennsylvania
Noam Abrams, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania
Bud Campbell, Grade 8, Albright Middle School, Houston, Texas
Mathew Steadman, Grade 8, Odle Middle School, Bellevue, Washington
Alex Chen, Grade 7, Odle Middle School, Bellevue, Washington
Xiaochang, Grade 8, Odle Middle School, Bellevue, Washington
Jeff Wong, Grade 8, St Ann, Wollaston, Massachusetts
Jennifer Liang, Grade 8, Odle Middle School, Bellevue, Washington
C.J. Walthall, Grade 8, Albright Middle School, Houston, Texas
Helen Wong, Grade 7, Odle Middle School, Bellevue, Washington
Scott Lemmon, Grade 9, East Mecklenburg High School, Charlotte, North Carolina
Ashley Monroe, Grade 9, Casady School, Oklahoma City, Oklahoma
Denny Chao, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Libbie Gies, Grade 9, Highland Park Senior High School, St. Paul, Minnesota
Marvin Scroggins, Grade 9, Highland Park Senior High School, St. Paul, Minnesota
Thuy Nguyen, Grade 11, Highland Park Senior High School, St. Paul, Minnesota
Alison Falkenhagen, Grade 9, Highland Park Senior High School, St. Paul, Minnesota
Wassia Khaja, Grade ,
Alison Miller, Grade 6, homeschooled, Niskayuna, New York
Tim Peterson, Grade , homeschooled, Rochester, New York
Jason Chiu, Grade 9, Laramie Junior High School, Laramie, Wyoming
Abbie Grier, Grade , Salisbury School, Salisbury, Maryland
Khari Taustin, Grade , Salisbury School, Salisbury, Maryland
Emily Buzicky, Grade 9, Highland Park Senior High School, St. Paul, Minnesota
Hyun Joo Lee, Grade 10, Ethel Walker School, Simsbury, Connecticut
Corey Boyd, Grade 9, North Pole High School, North Pole, Alaska
Kaitlin Primavera, Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania
Zimran Douglas, Grade 11, Wingate High School, Brooklyn, New York
Chris Perdue, Grade , Salisbury School, Salisbury, Maryland
Jane Milton and Sara Fitzsimmons, Grade 10, Mount Saint Joesph Academy, Flourtown, Pennsylvania
Alex Chernyavsky, Grade , Akiba Hebrew Academy, Merion, Pennsylvania
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