Some patterns of shaded and unshaded small triangles is given. The unshaded triangles are those which have at least one side on the edge of the big triangle. All of the other small triangles are shaded. The table below shows the number of small triangles.
- Find the total number of triangles in Pattern 55.
- Find the number of shaded triangles in Pattern 45.
- Find the number of unshaded triangles in Pattern 43.
| Pattern |
1 |
2 |
3 |
| Number of unshaded triangles |
3 |
6 |
9 |
| Number of shaded triangles |
1 |
3 |
7 |
| Total number of triangles |
4 |
9 |
16 |
(a)
Pattern for the total number of triangles
Pattern 1: (1 + 1)
2 = 4
Pattern 2: (2 + 1)
2 = 9
Pattern 3: (3 + 1)
2 = 16
Total number of triangles = (Pattern number + 1)
2 Total number of triangles in Pattern 55
= (55 + 1)
2 = 56
2 = 3136
(b)
Pattern for the number of shaded triangles
Pattern 1: 1 x (1 - 1) + 1 = 1
Pattern 2: 2 x (2 - 1) + 1 = 3
Pattern 3: 3 x (3 - 1) + 1 = 7
Number of shaded triangles = Pattern number x (Pattern number - 1) + 1
Number of shaded triangles in Pattern 45
= 45 x (45 - 1) + 1
= 45 x 44 + 1
= 1981
(c)
Pattern for the number of unshaded triangles
Pattern 1: 1 x 3 = 3
Pattern 2: 2 x 3 = 6
Pattern 3: 3 x 3 = 9
Number of unshaded triangles = Pattern number x 3
Number of unshaded triangles in Pattern 43
= 43 x 3
= 129
Answer(s): (a) 3136; (b) 1981; (c) 129
