The graph shows 3 figures formed by shaded and white triangles.
Pattern number |
1 |
2 |
3 |
... |
9 |
Number of shaded triangles |
0 |
2 |
5 |
... |
(i) |
Total number of triangles |
1 |
3 |
8 |
... |
(ii) |
- Complete the column for Figure 9. Give your answers for (i) and (ii) in numbers. (Eg 1, 2)
- What is the total number of triangles in Figure 19?
- How many shaded triangles are there in Figure 21?
(a)
Pattern for the number of shaded triangles
Figure 1: 0 =
0 =
1 - 1Figure 2: 0 + 2 =
2 =
1 + 2 - 1Figure 3: 0 + 2 + 3 =
5 =
1 + 2 + 3 - 1Formula:
Number of shaded triangles =
Sum of all numbers to Figure number - 1Number of shaded triangles in Figure 9
= (1 + 2 + 3 + ... + 9) - 1
= (9 x 10 ÷ 2) - 1
= 45 - 1
= 44 (i)
Pattern for the number of triangles from Figure 2 onwards.
Figure 2:
22 - 1 = 2 x 2 - 1 =
3 Figure 3:
32 - 1 = 3 x 3 - 1 =
8 Formula:
Number of triangles =
Figure number2 - 1 Total number of triangles in Figure 9
= 9
2 - 1
= 9 x 9 - 1
= 81 - 1
= 80 (ii)
(b)
Total number of triangles in Figure 19
= 19
2 - 1
= 19 x 19 - 1
= 361 - 1
= 360
(c)
Number of shaded triangles in Figure 21
= (1 + 2 + 3 + ... + 21) - 1
= (21 x 22 ÷ 2) - 1
= 231 - 1
= 230
Answer(s): (a) 44, 80; (b) 360; (c) 230