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The Math Forum @ Drexel University http://mathforum.org/pows/print/submissions.htm?publication=... Math Forum - Problem of the Week Submissions for Triple Tango 135 threads found, displaying 1 to 50.[First/Prev] 1, 2, 3 [Next/Last] Created Thread History Short Answer Long Answer Submitters School 11/25/2002 The following trieangles go together triangle 4 and triangle 2, triangles 3 and 6, and triangles 1 and 5. What I did was I would look for common lengths in the triangles, and if i found one i would add the angles near together, and if one of the angles equalled 180 degrees they were the right matches. I continued this process until i found three pairs. 79260 [n/a] 11/25/2002 Triangles 1 & 5 are a pair, triangles 6 & 3 are a pair, & triangles 2 & 4 are a pair. Triangles 1 & 5 are a pair, triangles 6 & 3 are a pair, & triangles 2 & 4 are a pair. First, I matched up the sides of the triangles with equal sides. Then, those sides on one angle must make a 180 degree angle, a strait line. Then, to check my work, I added together all of the sides of the triangle and checked that they equaled 180 degrees. 79033 [n/a] 11/25/2002 Triangle 1 goes with triangle To solve this problem, I just needed to know some basic geometry rules. One, is that a triangle has angles that equal 180 degrees. 5. Triangle 4 goes with Two, that you need congruent sides in order to not overlap. triangle 2. Triangle 3 goes with triangle 6. First, you want congruent sides from two triangles. For example: Triangle 1 has a side with a measure of 4.5 and so does triangle 5. Next, you have t 78340 2010 11/25/2002 Triangle 1 and 5, 2 and 4, 9 I cut the papers out, and then matched the sides! and 3 each pair up to form a bigger triangle 79335 8713 11/25/2002 You can put triangle 1 and 5 Well the process I used to solve this problem was that I looked for congruent sides on which you could combine 2 triangles without together at sides LG and letting them overlap. AB; you can put together triangles 3 and 6 at side HG and side XY; and you can put together triangle 2 and 4 by sides MO and RQ. 79395 [n/a] 11/25/2002 This page is very nice. i cut up the pieces and arranged do that they looked like they fit together good. 79397 793 11/25/2002 The three triangles i formed were by combining triangle one and triangle 5, triangle 2 and 4, and triangle 6 and 3. In order to find out which triangles go together, i considered what you need to make a triangle; 3 sides and interior angles that add up to 180 degrees. It took me a while to find out which triangles would go together, but i started out finding two sides that were of equal length, and randomly put them together and added up all three of the new corners to see if they add up to 180. Since triangle one's 4.5 side was the only side that matched with another measure on only trianle 5, i tried those first and they worked, when i put line seg LJ against line seg BC, then added up the angles in triangle A(CL)K it added up to 180 degrees. I did this using the side on triangle 6 of 3.4 to find that when matching line seg XY with HG, triangle ZP (GY)'s angles added up to 180 degrees. THat left me with triangles 2 and 4. I took the angles from triangle 4 and subracted each from 180 to see if the answer matched any of the angles on triangle 2. Since to make two of the angles connect to form a straight line they must add up to 180. I found that angle M plus angle Q equaled 180 degrees. So i added up the rest of the angles to find out tha when you make triagle NRO All the angles add up to 180. 78268 3004 A general descrition for the connection of triangle STU and DEF is two of the angles must be put together to equal a straight line of 180 degrees, then, the newly formed triangle must have only 3 sides and 3 angles...the angles must add up to 180 degrees by def. of the sum of the agles of a triangle. 1 of 6 11/26/2002 (1,5)/ (3,6)/ (2,4) The sides of the triangles that are going to be put together have to be congruent and one if the angles of the triangle has to add up to 180 degress when they are added to make a straight line (180 degrees), so that the angles of the triangle all together will equal 180 and be a true triangle. 79089 [n/a] 11/26/2002 The answer to "Triple Tango" is the following. Triangle #1 with #5, triangle #3 with #6 and triangle #2 with #4. To figure out which two triangles make a larger triangle, you measure the degrees of the triangle. The total of the three angles for the larger triangle must equal to 180 degrees. Also, any two adjacent triangles making up a straight line must equal to 180 degrees. For example, when #1 and #5 are joined at the 4.5" side, 79050 17298 11/24/10 1:22 PM The Math Forum @ Drexel University Created 11/26/2002 Thread History http://mathforum.org/pows/print/submissions.htm?publication=... Short Answer Triangle ABC and triangle JKL by AB to LJ, triangle PQR to triangle OMN by OM to RQ, triangle XYZ and triangle HGI by WY to HG. Long Answer A line that divides a triangle into two triangles will create a pair of supplementary angles. Using this logic, I found pairs of supplementary angles and adjacent corresponding congruent sides since (the line divison between two triangles to create a larger triangle). Submitters School 78872 1448 79405 [n/a] Triangles DEF and STU will have a pair of supplementary angles and at least one adjacent congruent side in order to form a larger triangle when put together. 11/26/2002 piss balls ass toldo totti shit 1. christmas porn dick-o clown we all make very good porn and a dick came to jimmy's ass and laid some creepy faget shit hawk porn 2. Bite my shit kill my clit and call me a ball tongue 3. eat my ass 11/26/2002 i think 3 and 4 look good together who cares they fit real nice 79406 [n/a] 11/26/2002 The final answer is 5. I used guess and check. 79407 [n/a] 11/26/2002 The three pairs of triangle that can be put together to form another, larger tringle are triangles 1 and 5, triangles 2 and 4, and triangles 3 and 6. The two triangles you are trying to combine to form a larger one must have lines with equal lengths that combine to become a shared side. Also, one angle from both of the triangles must be able to combine with the other to make a linear pair, two angles that form a straight line(180 degrees), so that they can form a straight line. This prevents another vertex from producing, making the figure into a quadrilateral. To find the answer to the problem, I found all the possible pairs of triangles that can form another, and picked out three that had no numbers twice in all of them. 79066 [n/a] 11/26/2002 Triangles 1 and 5, 6 and 3, and 2 and 4 are the pairs of triangles. You would look for two angles that equal 180 degrees and then see if the sides that make up the angle match upor are equal to each other. First thing that I did was found triangles that had sides that matched up. Then I looked at the angles in the triangles and added them to other angles to get 180 degrees. Then by trial and error I matched the triangles together. 77917 15764 11/26/2002 jojojojojojo lasdkjlas-20395-23094as;dlkma;sldf, 79067 [n/a] 11/26/2002 I think the answer is 30 degrees. This is what i got when i added them up. 78602 3508 11/26/2002 The three triangles are 1 and 5, 3 and 6, and 2 and 4. By using def and stu that is what i found when solving the problem. 77370 3508 11/26/2002 Triangles 1 and 5, triangles 3 and 6, and triangle 2 and 4 can all be put together to form Triangles 1 and 5 both share the similarity of the angle measurement of 4.5 (Triangle 1 = L to J and Triangle 5 = B to A) . traingles 3 and 6 both share the angle measurement of 3.4 (Triangle 3 = g to h, and Triangle 6 = x to y). Triangles 2 and 4 both share the angle measurement of 6.5 (Triangle 2= line M to 0 and Triangle 4= line RQ.) 78817 [n/a] 11/26/2002 To solve this problem, I used a spreadsheet. On this spreadsheet, I Triangles paired are classified triangles according to their similarities. triangles 1 and 5, triangles For classifing their similarities, the triangles had to have equal 2 and 4, and triangles 3 and sides, and one pair of adjacent angles that joined had to add up to 180 degrees(making a straight line). 6. Therefore, I paired up the triangles that had equal sides, and one 78637 15764 77720 906 pair of adjacent angles added up to 180 degrees. For triangles DEF and STU to be put together to form a larger triangle, one of DEF's sides must be equal to one of STU's sides. Also, one pair of their adjacent angles would have to add up to 180 degrees.(To make a straight line.) 11/27/2002 2 of 6 The biggest "hint" to me was that in the problem description it Triangle Pairs: 2 and 4, 1 states that the traingles cannot overlap. They certainly couldn't and 5, 6 and 3. With two fall short of eachother either, therefore, it made the most sense arbituary triangles DEF and that the triangles would line up perfectly, aka equally 180 degrees and continuing the straight line. This meant that if one triangle was STU, to be put together acute I had to make sure to match it with a triangle that contained they must have two angles an obtuse angle and so on. I accomplished the actual discovery of that compliment eachother which angle fit where by simple guess and check. (that added together equal 180 degrees) thus forming a 11/24/10 1:22 PM The Math Forum @ Drexel University Created Thread History http://mathforum.org/pows/print/submissions.htm?publication=... Short Answer Long Answer Submitters School complete line for one side of the tr WE GUESSED. HARD TO DO WITHOUT MOVING AROUND THE ANGLES OF THE TRIANGLES. 11/27/2002 Traingles 2 and 4, Traiangles 5 and 1, Triangles 3 and 6. 11/27/2002 the answer would be having the sides would have to fit together so the sides were equal so they would be the same lenghth so they were like 8.3" or the sides put togeather so something like that. they were equal so they would have to be the same length 11/27/2002 1,5 AND 2,4 AND 3,6 ARE THE REQUIRED PAIRS PAIR: 1&5 79419 [n/a] 77675 77689 16783 79420 [n/a] AB can be superimposed on LJ since they are of same length(4.5) angle KJL and angle ABC are linear pairs. ( since 114+66= 180) pair: 2&6 OM and RQ can be superimposed to form tiangle PON angles OMN AND RQP are linear pairs (since 119+61=180) pair: 3&4 HG and XY can be superimposed to form triangle IZG angles IHG and YXZ are linear pairs(since 90+90=180) 11/27/2002 the pairs are 1-5, 2-4, 3-6 When we look at the length of sides triangle 5 must be pair with triangle 1 because it's the only one that has the side witch can go together without overlapping. It's same with 2-4 too, and the remainin 3-6 should be the last pair. 77988 [n/a] 11/27/2002 no clue no clue 79421 [n/a] 11/27/2002 junk 1 and 5 went together, 3 and six went together and 4 and 2 went together. 79422 390 11/27/2002 triangle 1 and 5, 2 and 4, 3 i looked for equal sides between the triangles and for complementry angles. they wernt hard to find. in order to have a pair of triangles and 6 such as DEF and STU, that fit the qualifications, each one needs to 78753 906 77584 [n/a] have one side equal to another and an angle complementing the other triangle's angle. To sort the triangles into three groups you must do several things. First you must find two compatible triangles in which one side in each triangle are equal length. The two triangles must be congruent and linear. Meaning that one angle on each triangle must add up to 180 degrees or are supplementary. In this way the two triangles can be put together to form a larger triangle. The three pairs of triangles that I found were: Triangles 1 and 5, 2 and 4, and 3 and 6 11/27/2002 The three pairs of triangles that I found were: Triangles 1 and 5, 2 and 4, and 3 and 6 11/27/2002 The three pairs of triangles it was simple yopu just move them around untill you get to one that matches that can each be put together to form a larger triangle are triangle 6 and 3, triangles 1 and 5, and triangle 2 and 4. 77979 [n/a] 11/27/2002 The sides that are pairs because they have the same angles on both figures 1 and 5 are a pair, sides figures 4 & 2 are a pair, and a figures 6 & 3 78810 [n/a] 11/28/2002 4 with 2 6 with 3 1 with 5 4 with 2 >>>>> Since >>>> Since >>>> Since 78249 [n/a] 11/29/2002 1 and 5, 2 and 4, 3 and 6 I looked at the angle to see if it's the same or not. 79431 [n/a] 11/30/2002 The triangles are : 1 and 5, If you want to get three groups of triangles and two triangles in each group, you need to have two requirements. First, you need to 2 and 4, 3 and 6 have linear angles. And second, the two sides, one from each 78287 [n/a] triangle, needs to have congruent measurements. You might need to flip the triangles around or else it might not work. This is how you figure it out: Triangles 1 and 5 have a congruent side of 4.5 and 4.5, and you notice that you have to flip it in order to get the angles of 114 and 66 degrees to form linear angles. Linear angles are angles that share a side and lie on a line and they will add up to 180 degrees. So you flip it and rotate it until the two equal sides join the two triangles, and you see that they will make a bigger triangle. They are considered a group. Next, there are triangles 2 and 4. You would also have to flip and rotate it until it joins the two triangles together to form 3 of 6 11/24/10 1:22 PM The Math Forum @ Drexel University Created Thread History http://mathforum.org/pows/print/submissions.htm?publication=... Short Answer Long Answer Submitters School a bigger triangle. This time the two sides of the triangle have the measurements of 6.5 and 6.5. The two angles that are to form a linear angle have the measurements of 61 and 119 degrees. They will equal 180 degrees. The two triangles will form a group. Finally, there are the triangles 3 and 6. They will join together to form a bigger triangle like the rest. This is the last group of them all. This time the measurements for the two sides are 3.4 and 3.4. The two congruent angles are 90-degree angles. You will have to flip the triangle and rotate it to get the two 90-degree angles to form a linear angle. The sides will join and you will have a new triangle. 11/30/2002 6, 3, and 2, are one Triangle. 4, 1, and 5 are another triangle. For 6, 3, and 2, 78780 [n/a] 11/30/2002 The three pairs are GHI and XYZ, MNO and PQR, and LJK and ABC. Arbitrary triangles DEF and STU can be combined to make a single triangle if they (1) have at least one pair of congruent segments, and (2) if a linear pair is created by two of the angles Finding the answer to this problem is not so much about finding additional information to solve it as it is about sifting through the information that is already there. To find which triangles fit together properly and which do not, all I have to do is use the process of elimination and find which angles and sides match up. 78691 [n/a] LJK and ABC First, I will find LJK's counterpart. I can eliminate all choices except MNO and ABC, because none of the choices other then them have any congruent sides. It is impossible to match up two with noncongruent sides, because none of the sides would fit perfectly together, and so part of one side would rise up higher than the rest. The resulting figure would always be a quadrilateral, or worse. Next, I can eliminate MNO, because none of MNO's angles (that is, those which segment MN helps form) make a linear pair with LJK's angles (that JK helps form). If there are no linear pairs, then the four points M, N, J, and L cannot be colinear, and do not form a single segment. The combination of MNO and LJK would something like this: K/O - - - J/N- - L M Angle ABC forms a linear pair with angle KJL, so JKL and ABC can be combined to form one larger triangle. MNO and PQR First of all, MO and RQ are congruent, so the triangles can be matched up exactly. Second, angle NMO and angle RQP form a linear pair, and they are both partly formed by one of the two congruent segments. This means that N, M, Q, and P are all colinear. Therefore, a larger triangle is formed by segments NP, N(O/R), and (O/R)P. It looks somewhat like this: O/R - - - --------------N M/Q P GHI and XYZ Since I already made two pairs, only GHI and XYZ are left, so they are probably a pair also. Just to make sure, though, I will go through the same steps I did last time. GH and XY are congruent, so the triangles can be fitted together properly. Angle GHI and angle YXZ are a linear pair, so I, H, X, and Z are all colinear. Therefore, GHI and XYZ can be combined properly to form one triangle, like this: G/Y - - - -----------I H/X Z DEF and STU Two randomly selected triangles, DEF and STU, can be combined to form a single triangle if two conditions are met. First, DEF and STU must have two congruent sides. Second, a linear pair must be created by two of the angles that are partly made by those congruent segments. Of course, the two angles of that linear pair must be on separate triangles. Reflections When I came to the POW page to see the latest problem, I immediately realized that Triple Tango was, conceptually, extremely simple. Then, however, I realized the hard(er) part, which was figuring out how to 4 of 6 11/24/10 1:22 PM The Math Forum @ Drexel University Created Thread History http://mathforum.org/pows/print/submissions.htm?publication=... Short Answer Long Answer Submitters School word this correctly and understandably. I think the clarity of my explanations has improved since last time, but I still need to work on developing my communication skills. The hardest part of my explanation was figuring out how to say 'the angles formed partly by segment so-and-so', which I'm sure can be expressed more easily than that. 11/30/2002 triangles 1 and 5. triangles 2 and 3. triangles 4 and 6 look good 79438 [n/a] 12/01/2002 The three pairs of triangles were GHI and XYZ, PQR and MNO, and ABC and JKL. The angles of a triangle must always equal 180 degrees so, the triangles I created have to equal 180 degrees also. The first pair of triangles that I worked with (GHI and XYZ) was connected by a common side length of 3.4"(inches). I added the outside angles of tthe combined triange (64 + 26 + 66 + 24=180) which equals 180 degrees. The second pair of triangles (PQR and MNO)also had a common side length of 6.5"(inches). I added the outside angles of the combined triangle (39+22+29+90=180) to equal 180. With this pair I had to flip one of the triangles over to get the shape of a triangle. The third pair of triangles (ABC and JKL) had sides of an equal length (4.5"). I matched the common lengthed sides and added the outside angles of the new triangle (47+67+37+29=180) to find the total of 180 degrees. I had to flip one of the triangles over to make the triangle shape in this combination as well. 77539 [n/a] DEF and STU must have a side that is the same length to create a triangle. The four outside angles must equal 180 degrees. There are four outside angles, because each of the individual triangles has one one angle remaining, then another angle that on the combined triangle has been made into a straight line, and a third angle that is combined with the other triangles angle. The two combined angles make angles 3 and 4 to be added. Two arbitrary triangles can be combined in this manner. 12/01/2002 b m 79392 [n/a] 12/01/2002 1. 6 and 2 will match dsd 78044 [n/a] 12/01/2002 2 and 6 goes together 1 and 5 goes together 3 and 4 goes together. i found the length of the triangle and put them together then added them together and divide and got the length of the two that i choosed. 79442 6587 12/01/2002 Out of the six triangles that are given, you can specify them into three triangles. The way you can do this operation is by looking at al the sides of the triangles, and all the angles there are. THe triangles that go togheather must have at leas The way i found these pairs are by looking adjacent sides that are congruent and looking for te angles that make a linear pair. Once you have found the sides and angles that would make one big triangle, you now hav a pair. 78428 [n/a] 12/01/2002 The triangles that go together to form three larger triangles are 6+3, 2+4, and 1+5. First, I tried to find things that each triangle had in common. I set up 6+3 at the sides XY and HG. They both equal 3.4 inches, and the sum of all the angles is 180 degrees- the angle degrees are 82, 64, and 24. 79447 [n/a] 78560 [n/a] Next, the angles 2+4 are a larger triangle. When attaching to MN to QP, the sum of the angles is 180. The sum of all angles of a triangle equals 180 degrees. The individual angle degrees of the triangle pair are 129, 29, and 22. The last triangle set is 1+5. When attaching LJ and AB, there are no overlapping lines- LJ and AB both equal 4.5 inches. Since there are no overlapping sides, and the polygon has only three sides, it is a triangle. 5 of 6 asdf 12/01/2002 asdf 12/01/2002 The pairs are 1 and 5 2 and I just cut them out and fit them together like a puzzle 3 and 6 and 4 79450 [n/a] 12/02/2002 Line LJ from triangle one is equal to line AB from triangle five. The triangles that go Line GH from triangle 3 is equal to line XY from triangle 6. Line MO together is as follows: 1&5, from triangle 2 is equal to line RQ from triangle 4. 3&6, and 2&4. 78389 [n/a] 12/02/2002 Triangle 3 and 6 will make a triagnle when put together, triangles 2 and 4 will make a triangle, and triangles 1 and 5 make a triangle. If a 79034 [n/a] triangles 3 and 6 make a triangle because side GE in triangle 3 is 3.4" and so is side XY in triangle 6, also angle H plus angle X equals 180. Triangle 2 and 4 do the same thing due to the fact that side OM (in triangle 2) and side RQ (in triangle 4) both equal 6.5". also angle Q is 119 degrees, while angle M is 61 degrees, when put together, they equal 180. Therefore, i reflected triangle 2 over line ON and connected it to triangle 4. In a similar way, triangles 11/24/10 1:22 PM The Math Forum @ Drexel University Created Thread History http://mathforum.org/pows/print/submissions.htm?publication=... Short Answer Long Answer Submitters School triangle DEF has side EF with the length of 2.3" and angle E is 72 degrees, and triangle STU has side TU wi 5 and 1 make a triangle when connected to eachother. in triangle 1, side LJ is 4.5" as is line AB in triangle 5. the measure of angle J is 114 and the measure of angle B is 66. When connected they equal 180. 12/02/2002 bkjbjb 241212 78916 [n/a] 12/02/2002 Because of matching sides and cooresponding angles, triangles 1 and 5, 2 and 4, and 3 and 6 match up to form 3 different triangles. Triangles 1 and 5 match up to form one triangle by lining up on both of their 4.5 sides. then the 66+144=180, a straight line. and also the outside angles- 67+29(the newly formed angle) +47+37= 180, the number of degrees in a triangle. 79343 [n/a] 78233 3576 Triangles 2 and 4 match up similarly, but on the 6.5 side. here 61 and 119=180 creating a straight line while 29+22+90+39=180 degrees, a triangle. Triangles 3 and 6 also match up. THey match up along the 3.4 side. the two 90 degree angles form a straight line and then 26+66+64+24=180, again a triangle. Generally speaking, in order for two triangles to match up, they first must have a matching side. in triangles STU and DEF one of the sides of STU must be the same legnth as that of DEF. Also in order for the triangles to match up two of the angles (one from each triangle) must add up to 180 degrees to form a straight line along the bottom of the triangle. So an angle on either end of the line that equals the other triangle must add up to 180 degrees. Once these add up, here the triangles will match up, if they do. there is one final stipulation, the angles must add up to 180 degrees, the exterior angles. the two angles on the other end of the equal side that was not used before + the two other angles(one from DEF, one from STU) that were not on either end of the equal legnthed sides, must equal 180 degrees total to be a triangle. So overall to add up to a triangle there must be an equal legnth from a side of each triangle, 2 angles on each side from different triangles that add up to 180 degrees to form a line, and finally all the exterior angles must add up to 180 degrees. 12/02/2002 1. I found that the three pairs are triangles ABC and LJK, triangles MNO and RQP, and triangles XYZ and HGI. 2. For triangles DEF and STU to match up and create a larger triangle without overlapping, both triangles must have at least one congruent 1. I found out the pairs of the triangles by first figuring out that if I wanted to only have three sides, two of the sides had to make a straight line when they matched up. That meant that those corresponding angles had to be supplementary. And I knew if I had two sides that were not congruent which were matched up, an additional side would be created, making the resulting shape a parallelogram, not a triangle. 2. I knew that if there had to be two supplementary angles, or two right angles, and two congruent sides for the triangles to match up correctly, I knew all i needed was for triangles DEF and STU to both be right triangles with at least one side congruent. 135 threads found, displaying 1 to 50.[First/Prev] 1, 2, 3 [Next/Last] Just showing SUBMITTED threads Show threads of any status Number of results per page: 15, 50, 100, VIEW ALL Printer-friendly View © 1994-2010 Drexel University 6 of 6 http://mathforum.org/pows/ 11/24/10 1:22 PM