A series of patterns of shaded and unshaded small triangles is shown.
- Which pattern has 105 shaded triangles?
- Find the number of unshaded triangles in Pattern 27.
- Find the total number of triangles in Pattern 43.
| Pattern |
Number of
shaded triangles |
Number of
unshaded triangles |
Total number of
triangles |
| 1 |
1 |
15 |
16 |
| 2 |
4 |
21 |
25 |
| 3 |
9 |
27 |
36 |
| 4 |
16 |
33 |
49 |
(a)
Patterns for the number of shaded triangles
Pattern 1: 1 x 6 + 9 = 15
Pattern 2: 2 x 6 + 9 = 21
Pattern 3: 3 x 6 + 9 = 27
Pattern 4: 4 x 6 + 9 = 33
Number of shaded triangles = Pattern number x 6 + 9
Pattern number = (Number of shaded triangles - 9) ÷ 6
Pattern number for 105 shaded triangles
= (105 - 9) ÷ 6
= 96 ÷ 6
= 16
(b)
Pattern for the number of unshaded triangles
Pattern 1: 1
2 = 1
Pattern 2: 2
2 = 4
Pattern 3: 3
2 = 9
Pattern 4: 4
2 = 16
Number of shaded triangles in Pattern 27
= 27
2 = 729
(c)
Pattern for the total number of triangles
Pattern 1: (1 + 3)
2 = 16
Pattern 2: (2 + 3)
2 = 25
Pattern 3: (3 + 3)
2 = 36
Pattern 4: (4 + 3)
2 = 49
Total number of triangles = (Pattern number + 3)
2 Total number of triangles for Pattern 43
= (43 + 3)
2 = 46
2 = 2116
Answer(s): (a) 16; (b) 729; (c) 2116
