The following is made up of identical triangles. Study the pattern carefully.

1. Complete the table for Figure 6. Find the values of (i) and (ii). Give your answers in the following format. (Eg 1, 2)
2. A figure number in the pattern has a total of 121 triangles. What is the figure number?

(a)
Number of shaded triangles
Figure 1: 2 = 2
Figure 2: 5 = 2 + 3
Figure 3: 9 = 2 + 3 + 4

Formula:
Number of shaded triangles = [¹₂ x (Figure Number + 1) x (Figure Number + 2)] - 1

Number of unshaded triangles
Figure 1: 2 = 1 + 1
Figure 2: 4 = 1 + 1 + 2
Figure 3: 7 = 1 + 1 + 2 + 3

Formula: 
Number of unshaded triangles = [¹₂ x (Figure Number) x (Figure Number + 2)] + 1

Number of shaded triangles in Figure 6
= [¹₂ x (Figure Number + 1) x (Figure Number + 2)] - 1
= [¹₂ x (6 + 1) x (6 + 2)] - 1
= [¹₂ x 7 x 8] - 1
= 28 - 1
= 27

Number of unshaded triangles in Figure 6
= [¹₂ x Figure Number x (Figure Number + 1)] + 1
= [¹₂ x 6 x 7] + 1
= 21 + 1
= 22

(b)
Total number of triangles
Figure 1:  4 = 2 x 2
Figure 2:  9 = 3 x 3
Figure 3: 16 = 4 x 4

Formula:
Total number of triangles = (Figure Number + 1) x (Figure Number + 1)

(Figure Number + 1) x (Figure Number + 1) = 121
Figure Number + 1 = √121
Figure Number + 1 = 11

Figure Number = 11 - 1 = 10

Answer(s): (a) 27, 22; (b) 10