The diagram shows a sequence of patterns formed by identical triangles.
- Find the values of (i), (ii) and (ii). Give your answers in the following format. (Eg 1, 2, 3)
- A figure in the pattern has a total of 81 triangles. What is the Figure Number?
- Another figure has 41 more shaded triangles than unshaded triangles. What is the total number of triangles in this figure?
(a)
Number of shaded triangles:
Figure 1: 1 = 1
Figure 2: 3 = 1 + 2
Figure 3: 6 = 1 + 2 + 3
Formula:
Number of shaded triangles =
12 x Figure Number x (Figure Number + 1)
Number of unshaded triangles:
Figure 1: 0
Figure 2: 1 = 1
Figure 3: 3 = 1 + 2
Formula:
Number of unshaded triangles =
12 x (Figure Number - 1) x (Figure Number)
Total number of triangles:
Figure 1: 1 = 1 x 1
Figure 2: 4 = 2 x 2
Figure 3: 9 = 3 x 3
Formula:
Total number of triangles = Figure Number x Figure Number
Number of shaded triangles in Figure 5
=
12 x Figure Number x (Figure Number + 1)
=
12 x 5 x 6
= 15
Number of unshaded triangles in Figure 5
=
12 x (Figure Number - 1) x (Figure Number)
=
12 x 4 x 5
= 10
Total number of triangles in Figure 5
= 5 x 5
= 25
(b)
Figure Number that has a total of 81 triangles
= √81
= 9
(c)
Number of more shaded triangles than unshaded triangles:
Pattern 1: 1 - 0 = 1
Pattern 2: 3 - 1 = 2
Pattern 3: 6 - 3 = 3
Formula:
Number of more shaded triangles than unshaded triangles = Figure Number
Since there is 41 more shaded triangles than unshaded triangles, it is Figure Number 41.
Total number of squares in Figure Number 41
= 41 x 41
= 1681
Answer(s): (a) 15, 10, 25; (b) 9; (c) 1681