Certainly! Let's address each part of the question step by step.
### 1.1. Draw the next two triangles:
We start with the three match-stick triangle and build upon it.
- For the first triangle (simple triangle), you'll use 3 match-sticks.
Next two triangles (joined by one side):
- The second triangle shares one side with the first triangle, so it uses 2 more match-sticks.
- The third triangle shares one side with the second triangle, so it uses another 2 match-sticks.
Visually:
```
Δ Δ Δ
(3) + (2) + (2)
```
### 1.2. How many matches do Teddy use for the first triangle?
For the first triangle, Teddy uses 3 match-sticks.
### 1.3. How many matches do Teddy add for each triangle?
Each subsequent triangle shares one match-stick with the previous triangle and requires 2 additional match-sticks.
### 1.4. How many matches do Teddy need for the 6th triangle?
We need to calculate the total number of match-sticks for the 6th triangle.
The relationship is as follows:
- First triangle: 3 match-sticks
- Each additional triangle after the first: 2 more match-sticks
For the nth triangle, the total number of match-sticks can be calculated by the formula:
[tex]\[ \text{Total match-sticks} = 3 + 2 \times (n - 1) \][/tex]
Here [tex]\( n = 6 \)[/tex]:
[tex]\[ \text{Total match-sticks} = 3 + 2 \times (6 - 1) \][/tex]
[tex]\[ \text{Total match-sticks} = 3 + 2 \times 5 \][/tex]
[tex]\[ \text{Total match-sticks} = 3 + 10 \][/tex]
[tex]\[ \text{Total match-sticks} = 13 \][/tex]
So, Teddy will need 13 match-sticks for the 6th triangle.
### Conclusion
- For the first triangle, Teddy uses 3 match-sticks.
- For each additional triangle, 2 match-sticks are added.
- For the 6th triangle, Teddy needs a total of 13 match-sticks.
If you have any further questions or need clarification, feel free to ask!
