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Solutions for Session 2, Part B
See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | B9
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Problem B1 | |
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Here is the table filled in for the triangles with given lengths and for other sample triangles.
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Side A |
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Side B |
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Side C |
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Is it a triangle? |
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Can it be deformed? |
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4 |
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4 |
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4 |
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Yes |
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No |
4 |
3 |
2 |
Yes |
No |
3 |
2 |
1 |
No |
N/A |
4 |
3 |
2 |
Yes |
No |
1 |
2 |
4 |
No |
N/A |
2 |
4 |
4 |
Yes |
No |
3 |
1 |
1 |
No |
N/A |
2 |
3 |
3 |
Yes |
No |
2 |
4 |
2 |
No |
N/A |
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Other answers will vary individually, but no triangle will be deformable.
<< back to Problem B1
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Problem B2 | |
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No, this is not possible. If we attach the two sides of lengths of 4 units to the endpoints of the side of length 10, the first two sides will not meet at a point to create a triangle. Together they are too short.
<< back to Problem B2
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Problem B3 | |
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Three lengths can form a triangle only if the sum of the lengths of any two sides is greater than the length of the third side.
<< back to Problem B3
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Problem B4 | |
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Yes. Because the sum of lengths of any two sides is greater than the length of the remaining side, the two sides will be able to meet at a point and create a triangle when attached to the endpoints of the third side.
<< back to Problem B4
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Problem B5 | |
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No, three fixed lengths determine one and only one triangle. This is demonstrated by the fact that none of the triangles found in Problem B1 can be "deformed" into a different shape.
<< back to Problem B5
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Problem B6 | |
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Here is the table filled in for the quadrilaterals with given lengths and other sample quadrilaterals.
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Side A |
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Side B |
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Side C |
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Side D |
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Is it a quadri-
lateral? |
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Can it be deformed? |
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4 |
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4 |
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4 |
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4 |
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Yes |
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Yes |
4 |
3 |
2 |
2 |
Yes |
Yes |
3 |
2 |
1 |
1 |
Yes |
Yes |
4 |
1 |
2 |
1 |
No |
N/A |
1 |
1 |
1 |
4 |
No |
N/A |
2 |
2 |
2 |
2 |
Yes |
Yes |
1 |
4 |
3 |
1 |
Yes |
Yes |
1 |
3 |
3 |
4 |
Yes |
Yes |
2 |
3 |
4 |
1 |
Yes |
Yes |
4 |
1 |
1 |
2 |
No |
N/A |
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Other answers will vary individually, but all quadrilaterals will be deformable.
<< back to Problem B6
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Problem B7 | |
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As long as no more than two sides of a quadrilateral are equal in length, we can reorder the way the sides are connected and obtain a different quadrilateral. This is not the case with triangles: If we reorder the sides, we get the same triangle.
<< back to Problem B7
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Problem B8 | |
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Four lengths can form a quadrilateral as long as the sum of the lengths of any three sides is greater than the length of the fourth side.
<< back to Problem B8
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Problem B9 | |
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Yes. For example:
Also, if the sides are not the same length, ordering them differently will produce different quadrilaterals.
<< back to Problem B9
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