The following is made up of identical triangles. Study the pattern carefully.
- Complete the table for Figure 6. Find the values of (i) and (ii). Give your answers in the following format. (Eg 1, 2)
- A figure number in the pattern has a total of 144 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) = 144
Figure Number + 1 = √144
Figure Number + 1 = 12
Figure Number = 12 - 1 = 11
Answer(s): (a) 27, 22; (b) 11