The table below shows the number of triangles in each diagram. Diagram 1 2 3 4 5 6 7 8 9 10 Small triangles 1 4 9 16 25 36 Black triangle 1 3 5 7 9 11 Grey triangles 0 1 3 6 10 White triangles 0 0 1 3 6 10 2.5.1 Complete the column for diagrams 7 and 8.
Question: The table below shows the number of triangles in each diagram. Diagram 1 2 3 4 5 6 7 8 9 10 Small triangles 1 4 9 16 25 36 Black triangle 1 3 5 7 9 11 Grey triangles 0 1 3 6 10 White triangles 0 0 1 3 6 10 2.5.1 Complete the column for diagrams 7 and 8. (4)
The table below shows the number of triangles in each diagram. The diagrams are made of small black, grey and white triangles arranged in a square grid. The number of small triangles in each diagram is equal to the square of the side length of the grid. For example, diagram 1 has a side length of 1 and 1 small triangle, diagram 2 has a side length of 2 and 4 small triangles, and so on. The number of black triangles in each diagram is equal to the odd numbers from 1 to 11. For example, diagram 1 has 1 black triangle, diagram 2 has 3 black triangles, and so on. The number of grey triangles in each diagram is equal to the triangular numbers from 0 to 10. A triangular number is the sum of the natural numbers from 1 to n. For example, diagram 2 has 1 grey triangle (1), diagram 3 has 3 grey triangles (1+2), and so on. The number of white triangles in each diagram is also equal to the triangular numbers from 0 to 10, but shifted one position to the right. For example, diagram 3 has 1 white triangle (0), diagram 4 has 3 white triangles (1), and so on.
Diagram | Small triangles | Black triangle | Grey triangles | White triangles
---|---|---|---|---
1 | 1 | 1 | 0 | 0
2 | 4 | 3 | 1 | 0
3 | 9 | 5 | 3 | 1
4 | 16 | 7 | 6 | 3
5 | 25 | 9 | 10 | 6
6 | 36 |11|15|10
7|49|13|21|15
8|64|15|28|21
9|81|17|36|28
10|100|19|45|36
To complete the column for diagrams 7 and 8, we can use the patterns described above. Diagrams 7 and 8 have side lengths of 7 and 8 respectively, so the number of small triangles is equal to $7^2=49$ and $8^2=64$. The number of black triangles is equal to the next odd numbers after $11$, which are $13$ and $15$. The number of grey triangles is equal to the next triangular numbers after $15$, which are $21$ and $28$. The number of white triangles is equal to the same triangular numbers as the grey ones, but shifted one position to the right, which are $15$ and $21$.
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