The three diagrams show the highest number of intersections obtained from 2, 3 and 4 lines respectively.

1. What is the highest number of segments for 12 straight lines?
2. What is the maximum number of intersections obtained from 35 straight lines?
3. What is the maximum number of regions obtained by using 20 straight lines?

(a)
Pattern for number of segments
1 Line: 1² = 1
2 Lines: 2² = 4
3 Lines: 3² = 9
4 Lines: 4² = 16

Formula
Number of segments = Number of lines²

Number of segments for 12 Lines
= 12²
= 144

(b)
Pattern for the number of intersections
Line 1: 1 x (1 - 1) ÷ 2 = 0
Line 2: 2 x (2 - 1) ÷ 2 = 1
Line 3: 3 x (3 - 1) ÷ 2 = 3
Line 4: 4 x (4 - 1) ÷ 2 = 6

Formula
Number of intersection = Line number x (Line number - 1) ÷ 2

Number of intersection for 35 lines
= 35 x (35 - 1) ÷ 2
= 595

(c)
Pattern for the number of regions
Line 1: (1² + 1) ÷ 2 + 1 = 2
Line 2: (2² + 2) ÷ 2 + 1 = 4
Line 3: (3² + 3) ÷ 2 + 1 = 7
Line 4: (4² + 4) ÷ 2 + 1 = 11

Number of regions = (Number of line² + Number of line) ÷ 2 + 1

Number of regions using 20 lines
= (20² + 20) ÷ 2 + 1
= 420 ÷ 2 + 1
= 210 + 1
= 211

Answer(s): (a) 144; (b) 595; (c) 211