Tangram Pieces - December 4-8, 1995
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Here is a picture of a tangram, which is a puzzle that consists of a square cut into seven pieces. Which of the triangles are congruent to each other?
If AB = 1, what is the area of each of the seven pieces?
Annie says: Solutions
Good responses this week, but some of the incorrect solutions failed to say which triangles were congruent - make sure you read the problem carefully!
We don't need to know formulas for areas for this one, much less the specific edgelengths of the pieces. It is sufficient to carve up the square into nice pieces, which is what most of the students did.
Shannon Firth and Christina DiGregorio did a nice thing in their explanation. They say they drew 'auxiliary' lines to split up some of the larger pieces into small triangles, making that somewhat of a unit. They also pointed out that just because shapes have the same area it doesn't mean they are congruent.
A number of students showed that their answer was correct by explaining that the areas should add up to 1, which is the area of the whole square - excellent!
There were several new schools this week, including one in Israel, another one in Australia, and one in Stow, Ohio. Following are highlights, and the names of the people who submitted correct solutions and their solutions are also available. There was a great variety of responses in terms of how much was explained - we've got it all, from minimalist to journal-length articles! Make sure you check out the range of responses.
Shannon Firth, and Christina DeGregorio Grade 9 School: Mt. St. Joseph Academy, Flourtown, PA Triangles AEB and DEA are congruent to each other. Each has an area of .25 in. squared. We found the area of each triangle by dividing the square (1 in. squared) by four. We also found that triangles BHJ and FEI are congruent. Each has a measure of .0625. The three remaining pieces, the parallelogram DFIG, the parallelogram EHJI, and the triangle GJC all have a measure of .125; however, they are not congruent. We've already explained to you how we got the answer for the two large triangles AEB and DEA. For the five other pieces of the square, we drew auxiliary lines EJ, FG, and IC in triangle BDC. By doing this we formed eight congruent triangles. The purpose of this was merely to enable us see things more clearly. Triangle HJB is congruent to triangle EFI, simply because they are two of the eight congruent triangles. The other six triangles paired up and formed three different shapes. Even though their areas were equal, (because they were formed from two congruent triangles) their geometric shapes were not the same, so they were not congruent. Then we checked our answer: .125 (3) units squared .0625 (2) units squared +.25 (2) units squared _______ 1 unit squared
Cassie Gorish Grade: 8 School: Murray Junior High School 1. I used tangram pieces to find out that triangles ABE=AED and BHJ=EFI. 2. By mathematically dividing and comparing triangles, I found that the figures have the following areas: ABE = .25 AED = .25 BHJ = .0625 EFI = .0625 JGC = .125 FIGD = .125 HEIJ = .125 As a check, I added the areas together, and the sum equaled 1.
Stephanie Jablonski Grade: 8 School: Upper Twp. Middle School Tuckahoe, NJ Triangles ADE and ABE = 1/4. Parallelogram DFIG, triangle GJC and square EHJI are 1/8. Triangles HBJ and FEI are 1/16. I got these answers by first knowing that triangles ADE and ABE are 1/2 of the square together so each they are 1/4. Then I saw that I could make a parallelogram into a rectangle by cutting it up. I took the top of the parallelogram off so that the piece that was cut off was a triangle. Then I put the triangle on the bottom of the parallelogram to make a rectangle. Now I had a rectangle I tried fitting the rectangle into the original square which came out to fit 8 pieces. It was only one of those pieces so I knew that it was 1/8. Now I moved on to triangle GJC. I kept it the way it was and tried fitting as many as I could into the original triangle and I fit 8.I labeled it 1/8. Now I had triangles HBJ, FEI, and GJC and Square EHJI. Triangle GJC fit 8 times which was 1/8. The HBJ and FEI fit in GJC and square EHJI. The triangles that are congruent are triangles ABE and AED. The other two triangles that are congruent are triangles BHJ and FEI.
Thomas S. Kuo Grade: 7 School: Murray Junior High School, Ridgecrest, California Triangles EAB and EDA are congruent. Triangles HBJ and EIF are congruent. The areas of each piece are: EAB = EDA = 1/4, HBJ = EIF = 1/16, CGJ = 1/8, EHJI = 1/8, FIGD = 1/8. For triangle EAB and EDA - These two triangles are congruent and together take up 1/2 of the area of the square. So the area of each triangle is one half of 1/2 which is 1/4. For triangles HBJ and EIF - Triangle HBJ has the height as 1/4 and the base as 1/2. So the area is 1/2 * 1/2 * 1/4 = 1/16. For triangle CGJ - Triangles CGJ and CDB are similar. Their sides are at a ratio of 1/2, so their area has a ratio of 1/4. I also know the area of CDB one half of the square which is 1/2. I then solve equation x : 1/2 = 1/4 for x for the area of CGJ which is 1/2 * 1/4 = 1/8. For EHJI - Using the Pythagorean Theorem, I calculate the length of BD as sqrt(2). I also know the length of one side of the square is a quarter of sqrt(2) which is sqrt(2)/4. I then square it to find the area of this square, which is 1/8. For FIGD - I know the area of triangle CDB is 1/2 and it contains 5 pieces. The area of FIGD is 1/2 subtracts sum of areas of other four pieces, 1/16 + 1/16 + 1/8 + 1/8 = 3/8, and is 1/8.
Michael Li Grade 8 My answer for the 7 pieces was, triangles ABE and ABD each with an area of 1/4, triangles BHJ and EFI each w/ an area of 1/16, and Tri GCJ and Quads HEIJ and FDGI each w/ an area of 1/8. I found the area of triangles ABE and ADE by seeing that they were congruent and filled up half of the square. So if that large portion is 1/2 and the triangles are congruent, they both equal 1/4 in area. At the other half, BCD, if you bring line segment AI down to point C, triangle BCD will be cut into 1/2 or 1/4 pieces. On each 1/4 area piece there is a quadrilateral and each quadrilateral is double the size of each small triangle BHJ and EFI. So, it would have to equal 1/8 because 2 triangles are left and together they have to equal 1/8 for the triangle to equal 1/4. Now that we know the quadrilaterals equal 1/8, the triangles must be 1/16 because they are each exactly 1/2 of the quadrilaterals. Now, if you take out the extension of segment AI, triangle GJC will equal 2/16 or 1/8. So, the following are the areas of the shapes: Tri ABE and ADE = each 1/4 * 2 = 1/2 Quad HEIJ and FDGI = each 1/8 * 2 = 1/4 Tri GJC = 1/8 = 1/8 Tri BHJ and EFI = each 1/16 * 2 = 1/8 Total 1
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