1. How many sticks are needed to form Pattern 28?
2. In which pattern will each side of the triangle measure 20 cm?
3. Calculate the number of shaded triangles in Pattern 54.

(a)
Number of sticks:
Pattern 1: 1 x 3   = (1) x 3 = 3 sticks
Pattern 2: 3 x 3   = (1 + 2) x 3 = 9 sticks
Pattern 3: 6 x 3   = (1 + 2 + 3) x 3 = 18 sticks
Pattern 4: 10 x 3 = (1 + 2 + 3 + 4) x 3 = 30 sticks

Sum of numbers = Pattern Number x (Pattern Number + 1) ÷ 2

Formula:
Number of sticks = Sum of numbers up to Pattern Number x 3

Number of sticks needed for Pattern 28
= (1 + 2 + 3 + ... + 28) x 3
= [28 x (28 + 1) ÷ 2] x 3
= [28 x 29 ÷ 2] x 3
= 406 x 3
= 1218

(b)
Length of each side of the triangle:
Pattern 1: 1 x 2 = 2 cm
Pattern 2: 2 x 2 = 4 cm
Pattern 3: 3 x 2 = 6 cm
Pattern 4: 4 x 2 = 8 cm

Formula:
Length of each side of the triangle = Pattern Number x 2

Pattern Number with each side of the triangle measuring 20 cm
= 20 ÷ 2
= 10 

(c)
Pattern 1: Number of shaded triangles = 0
Pattern 2: Number of shaded triangles = 0 + 1
Pattern 3: Number of shaded triangles = 0 + 1 + 2
Pattern 4: Number of shaded triangles = 0 + 1 + 2 + 3

Pattern for number of shaded triangles = Sum of numbers up to (Pattern Number - 1)

Number of shaded triangles in Pattern 54
= [(Pattern Number - 1) x (Pattern Number)] ÷ 2
= (53 x 54) ÷ 2
= 1431

Answer(s): (a) 1218; (b) 10; (c) 1431