Alan is using matchsticks to make isosceles triangles. Isosceles triangles have at least two sides the same length.How many different single isosceles triangles can he make with 45 matchsticks, using all the matchsticks each time?
The answer is 11, but I can only work out 7, PLEASE HELP!!
1 X 3 M-sticks
1 x 5 '
1 x 6 '
2 x 7 '
2 x 8 '
X = # short sticks.
2X = # Long sticks.
x + 2x = 45
X = 15 Short sticks.
2X = 30 Long sticks.
Therefore, you can make 15 triangles
using 15 short sticks and 30 Long sticks.
If the answer was 11 triangles:
11triangles * 3sticks/triangle = 33 sticks.
To find all the different single isosceles triangles Alan can make with 45 matchsticks, here are the possible combinations:
1. Using 3 matchsticks:
- 1 triangle
2. Using 5 matchsticks:
- 1 triangle
3. Using 6 matchsticks:
- 1 triangle
4. Using 7 matchsticks:
- 1 triangle
5. Using 8 matchsticks:
- 1 triangle
So far, you have correctly found 5 different triangles. Let's continue:
6. Using 9 matchsticks:
- 1 triangle
7. Using 10 matchsticks:
- 1 triangle
8. Using 11 matchsticks:
- 1 triangle
9. Using 12 matchsticks:
- 1 triangle
10. Using 13 matchsticks:
- 1 triangle
11. Using 14 matchsticks:
- 1 triangle
Therefore, Alan can create a total of 11 different single isosceles triangles using all the 45 matchsticks.
To solve this problem, let's break it down step by step.
First, we need to determine the possible lengths for the equal sides of the isosceles triangles. Since Alan is using matchsticks and has a total of 45 matchsticks, we can assume that the individual matchstick length is 1 unit.
Now, let's consider the possible lengths for the equal sides. Let's start with the smallest possible length, which is 1 unit.
For a triangle with 1 unit length sides, we would need 3 matchsticks. However, we only have 1 unit length matchsticks, so we cannot form a triangle of this size.
Let's move on to the next possible length, which is 2 units.
For a triangle with 2 unit length sides, we would need 5 matchsticks. Since we have a total of 45 matchsticks, we can form a maximum of 45/5 = 9 triangles of this size. However, we need to ensure that we are using all the matchsticks each time.
To form a triangle with 2 unit length sides and using all 45 matchsticks, we can take the following approach:
- Start with the maximum number of triangles we can form, which is 9.
- Each triangle would use 5 matchsticks, so the total number of matchsticks used would be 9 * 5 = 45.
So, we have found one possibility for using all 45 matchsticks with triangles of 2 unit side lengths.
Let's move on to the next possible length, which is 3 units.
For a triangle with 3 unit length sides, we would need 6 matchsticks. Using a similar approach as before, we can calculate the maximum number of triangles we can form: 45/6 = 7.5.
However, since we need to use all the matchsticks each time, we can only form a maximum of 7 triangles with 3 unit side lengths.
We can follow this process for the remaining possible lengths.
For triangles with 7 and 8 unit side lengths, we can form 2 triangles each. (2 * 7 = 14 matchsticks and 2 * 8 = 16 matchsticks, both less than 45)
Summing up all the triangles:
- Triangles with 2 unit side lengths: 9 triangles.
- Triangles with 3 unit side lengths: 7 triangles.
- Triangles with 7 unit side lengths: 2 triangles.
- Triangles with 8 unit side lengths: 2 triangles.
Adding these numbers together, we have a total of 9 + 7 + 2 + 2 = 20 triangles.
So, based on the analysis above, it seems that there might be a discrepancy. The answer should be 20 triangles, not 11.
Please double-check your calculations, and if there are any additional rules or constraints that were not mentioned, please let us know so we can revisit the problem.