Bob the builder has many sticks of length 3, 5, and 7. He wants to form triangles whose edges consist of exactly 1 stick. How many non-congruent triangles can he form with the sticks?
Examine the unique triplets of numbers that can be created with 3,5, and 7. There are 10 of them. Any other triplet of numbers would be a rotation or reflection of these. 3−3−33−3−53−3−73−5−53−5−73−7−75−5−55−5−75−7−77−7−7 You might think that the answer is 10, but remember that in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This eliminates 3−3−7 as a possible triangle, so the answer is 9
The correct answer is: 9
Heck, the only set of sides which cannot work is
3,3,7
So, list all those with all 3 sides the same,
those with two sides the same
those with all different sides.
To determine the number of non-congruent triangles Bob can form with sticks of length 3, 5, and 7, we can use the triangle inequality. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's analyze the possible combinations of sticks Bob can use to form triangles:
1. Using the stick of length 3:
- Combining it with the stick of length 5: The sum of the lengths is 3 + 5 = 8, which is greater than the remaining stick of length 7. Therefore, Bob can form a triangle with sides of length 3, 5, and 7.
2. Using the stick of length 5:
- Combining it with the stick of length 3: The sum of the lengths is 5 + 3 = 8, which is greater than the remaining stick of length 7. Therefore, Bob can form another triangle with sides of length 3, 5, and 7.
3. Using the stick of length 7:
- Combining it with the stick of length 3: The sum of the lengths is 7 + 3 = 10, which is greater than the remaining stick of length 5. Therefore, Bob can form a triangle with sides of length 3, 5, and 7.
Note that these three triangles are non-congruent because they have different side lengths.
Therefore, Bob can form 3 non-congruent triangles using sticks of length 3, 5, and 7.