Question 1119203: A geometry teacher sent her students home with an assignment to analyze some lists of given side lengths—a,b, and c — to determine if each list of side lengths could form a triangle. And if a triangle was possible for any of the given lists, to draw a diagram of the triangle and label it with sides a, b, and c, and angles A, B, and C as shown below (side a opposite angle A, b opposite B, c and C opposite ).
Picture: https://api.agilixbuzz.com/Resz/~PW4LEAAAAAQsZBc1R9ciiB.CLm56DbZ6T1uAwIk1KSyJA/48780739,5F4,0,0/Assets/Media/Images/43.1-HOT2-Tris.jpg
The student was then asked to determine which angle was smallest and which was largest and to explain their reasoning. The teacher gave three lists to each student. Look at each student’s list,
Caroline got the following three lists:
List 1:
a = 6
b = 4
c = 5
List 2:
a = 10
b = 25
c = 13
List 3:
a = 6
b = 15
c = 17
Caroline’s answer looked like this:
Picture: https://api.agilixbuzz.com/Resz/~PW4LEAAAAAQsZBc1R9ciiB.CLm56DbZ6T1uAwIk1KSyJA/48780739,5F4,0,0/Assets/Media/Images/43.1-HOT2-Tris2.jpg
List 1: Angle A = largest; Angle C = smallest
The numbers in list 1 are all close enough that they can make a triangle. Angle A is the largest because it is across from the longest side, so the other two sides have to be spread wide enough to reach it. Angle C is the smallest because it is across from the smallest side so it can’t reach very far.
List 2: These numbers cannot make a triangle because 25 is too long and 10 & 13 can’t reach the ends of the 25 side.
List 3: Angle C = largest; Angle A = smallest
The lengths of this triangle are all close enough to make a triangle because 2 long sides can connect to a short side unlike List 2 where there were 2 short sides. Angle C is the largest because it is across from the longest side. Angle A is the smallest because it is across from the shortest side.
Graziella got the following three lists:
List 1:
a = 14
b = 2
c = 22
List 2:
a = 16
b = 5
c = 11
List 3:
a = 8
b = 9
c = 6
Graziella’s answer looked like this:
Picture: https://api.agilixbuzz.com/Resz/~PW4LEAAAAAQsZBc1R9ciiB.CLm56DbZ6T1uAwIk1KSyJA/48780739,5F4,0,0/Assets/Media/Images/43.1-HOT2-Tris3.jpg
List 1: No, c is too long 22. Side b is too short 2. It would not be a triangle.
List2: B has the smallest angle because b is the smallest side. A is the largest because a 16 is the longest side, so A has a wider angle.
List 3: This can be a triangle because of the length of each side. B has the largest angle because the opposite side b is longer, therefore, since is shorter, the angle should open more in order for b and c to touch. On the other hand, C has the smallest angle because c is shorter, therefore, C doesn’t open as wide as the other angles.
a. Critique the work of Graziella and Caroline. What did they do right? What did they do wrong?
b. What grade would you give Graziella and Caroline for this assignment? Why?
c. What clarifications can you add to their work to make their answers more worthy of a top grade?
Answer by solver91311(24713)   (Show Source):
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Caroline
The result for list 1 is completely incorrect. In the first place, side b is shorter than side c and therefore angle B is the smallest angle of the three. In the second place, her diagram represents a right triangle (from the little box in the corner of the largest angle). But with sides 6, 5, and 4, the triangle cannot be right because 36 is NOT equal to 25 plus 16. Thank you, Pythagoras. Her analysis that "...numbers in list 1 are all close enough..." is not a rigid enough argument to prove the case that a triangle can be made. What does "close enough" mean? These numbers are consecutive integers, but so are 1, 2, and 3 and these three side lengths cannot form a triangle.
Her conclusion for list 2 is correct, but she should not have drawn a diagram at all. The instructions say "If a triangle was possible...draw a diagram...". She failed to follow instructions and therefore should only get partial marks for this part of the assignment. Also, her language describing her analysis is decidedly imprecise.
For list three, her conclusion is correct, but her analysis is faulty. "2 long sides can connect to a short side" is an inappropriate criterion for determining whether a triangle can be formed. In every case, the sum of the two shortest sides must be compared to the longest side. If the sum of the two shortest sides is strictly greater than the longest side, then a triangle can be formed.
Graziella
The conclusions and the diagrams for all three lists are 100% correct. She followed instructions and did not attempt a diagram for list one where a triangle was not possible.
However, her analysis for list one is faulty. It is insufficient to note that "...c is too long 22. Side b is too short 2..." in order to prove that a triangle cannot exist. She should have demonstrated that the sum of the measures of sides a and b was less than the measure of side c. After all, had side a measured in the range 20 < a < 24, then a triangle would have been possible with c = 22 and b = 2.
Even though she concluded that triangles exist for lists two and three, she failed to provide an analysis that proves it for either of these two lists.
In General
I would give both assignments approximately the same mid-range grade, perhaps a C or 70+ percent. I would want the grade to reflect a reasonable level of intuitive competence, but also to indicate that there is a good deal of room for improvement -- especially if continued work in more advanced mathematics is in either of these two students' futures.
In order to receive top marks, I would expect the student's paper to begin with a statement of the Triangle Inequality Theorem, namely The sum of the lengths of any two sides of a triangle is greater than the length of the third side. From which we can conclude that the sum of the measures of the two shortest sides must be strictly greater than the measure of the longest side.
Then the analyses for each of the lists should be couched in parallel terms:
"The sum of the two smallest given numbers is ___ " Then either "This is smaller than (or possibly equal to) the largest number, therefore no triangle can be formed." OR "This is larger than the largest number and therefore a triangle can be formed. The largest side is ___, so the largest angle is ___. The smallest side is ___, so the smallest angle is ___." And in the latter case a diagram should be included.
John
My calculator said it, I believe it, that settles it
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