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 10 straight lines?
2. What is the maximum number of intersections obtained from 39 straight lines?
3. What is the maximum number of regions obtained by using 42 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 10 Lines
= 10²
= 100

(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 39 lines
= 39 x (39 - 1) ÷ 2
= 741

(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 42 lines
= (42² + 42) ÷ 2 + 1
= 1806 ÷ 2 + 1
= 903 + 1
= 904

Answer(s): (a) 100; (b) 741; (c) 904