The three diagrams show the highest number of intersections obtained from 2, 3 and 4 lines respectively.
- What is the highest number of segments for 18 straight lines?
- What is the maximum number of intersections obtained from 29 straight lines?
- What is the maximum number of regions obtained by using 50 straight lines?
(a)
Pattern for number of segments
1 Line: 1
2 = 1
2 Lines: 2
2 = 4
3 Lines: 3
2 = 9
4 Lines: 4
2 = 16
Formula
Number of segments = Number of lines
2 Number of segments for 18 Lines
= 18
2 = 324
(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 29 lines
= 29 x (29 - 1) ÷ 2
= 406
(c)
Pattern for the number of regions
Line 1: (1
2 + 1) ÷ 2 + 1 = 2
Line 2: (2
2 + 2) ÷ 2 + 1 = 4
Line 3: (3
2 + 3) ÷ 2 + 1 = 7
Line 4: (4
2 + 4) ÷ 2 + 1 = 11
Number of regions = (Number of line
2 + Number of line) ÷ 2 + 1
Number of regions using 50 lines
= (50
2 + 50) ÷ 2 + 1
= 2550 ÷ 2 + 1
= 1275 + 1
= 1276
Answer(s): (a) 324; (b) 406; (c) 1276