6.1 Line Segments

A linear relation is a relation between two variables that form a straight line when graphed. At times we will only be concerned with a portion of a linear relation. A portion of a linear relation is called a line segment. This lesson will cover how to find the length of a line segment because we will need to use this skill in later lessons.

To name a line segment, use the letters that represent the endpoints. For example, if a line segment spans from point A (1, 2) to point B (2, 7), we would call the line segment AB.

Vertical Line Segments

One method that can be used to find the length of a vertical line segment is to plot the points, connect the points with a straight line, and then count the number of spaces that the line spans. You will also explore a second method that can be used to find the length of a vertical line segment.

Practice 1

Plot the following points on the grids, and connect the points. Label each point with its designated letter. Determine the length of each line segment.

a) A (2, 4) and B (2, 1)

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6. and a scale of 1.

 b) C (–5, –1) and D (–5, –6)

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6. and a scale of 1.

Show how you could find the length of each line segment without plotting the points.

c) A (2, 4) and B (2, 1)

 

 d) C (–5, –1) and D (–5, –6)

 
Practice 2

Determine the length of the line segment formed by each pair of points.

a) A (3, 5) and B (3, 3)

 

 b) X (0, 3) and Y (0, –1)

 

 c) N (–12, –2) and M (–12, –15)

 

Horizontal Line Segments

One method that can be used to find the length of a horizontal line segment is to plot the points, connect the points with a straight line, and then count the number of spaces that the line spans. You will also explore a second method that can be used to find the length of a horizontal line segment. Practice on the grids below.

Practice 3

Plot the following points on the grids, and connect the points. Determine the length of each line segment.

a) X (1, 4) and Y (5, 4)

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6. and a scale of 1.

 b) P (–3, –2) and Q (5, –2)

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6. and a scale of 1.

Show how you could find the length of each line segment without plotting the points.

c) X (1, 4) and Y (5, 4)

 

 d) P (–3, –2) and Q (5, –2)

 
Practice 4

Determine the length of the line segment formed by each pair of points.

a) N (–6, 6) and O (–1, 6)

 

 b) F (5, –4) and G (10, –4)

 

 c) C (–20, –7) and D (20, –7)

 

Diagonal Line Segments

When finding the length of a diagonal line segment, we cannot use the methods above. When looking at a diagonal line, you will notice that it is difficult to count the spaces because the line spans portions of a space. To find the length of a diagonal line segment, you will need to use the Pythagorean theorem.

Table 6.1.1
The Pythagorean Theorem
A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.

Given the right triangle \Delta ABC, where c is the length of the hypotenuse and a and b are the lengths of the legs,

{a}^{2}+{b}^{2}={c}^{2}
Example 1

Determine the length of the line segment from A (2, 3) to B (7, 6). Write your answer as an exact value and as a decimal to the nearest tenth.

Table 6.1.2
Steps Solution
Plot the points on a grid. The graph shows the x y coordinate plane. The x -axis runs from 0 to 8. The y -axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6). An additional point is plotted at (7, 3). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (7, 3) and from (7, 3) to (7, 6) forming the legs.
Count the horizontal distance from one point to the other. Then count the vertical distance from one point to the other. Horizontal distance = 5 spaces
Vertical distance = 3 spaces
Use the Pythagorean theorem to find the length of the diagonal line segment. {a}^{2}+{b}^{2}={c}^{2}
{5}^{2}+{3}^{2}={c}^{2}
25+9={c}^{2}
34={c}^{2}
\(\sqrt{34} = \sqrt{{c}^{2}}
\(\sqrt{34} = c
5.83 = c
Practice 5

Plot the points, then determine the length of the line segment formed by the points. Write your answers as an exact value and as a decimal to the nearest tenth.

a) X (–5, –3) and Y (2, 4)

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6. and a scale of 1.

 b) V (–6, 5) and Y (3, –6)

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6. and a scale of 1.

Homework 

  1. Find the length of the line segment formed by each pair of points.

    a) M (–5, 7) and N (3, 7)

     

    		 b) P (4, 8) and Q (–2, 8)

     

    		 c) X (1, 10) and Y (–6, 10)

     
    d) A (–67, 12) and B (0, 12)

     

    		 e) C (–3, 5) and D (7, 5)

     

    		 f) E (90, 6) and F (–15, 6)

     

  2. Find the length of the line segment formed by each pair of points.

    a) L (4, 7) and K (4, –3)

     

    		 b) J (–2, 8) and I (–2, 5)

     

    		 c) H (10, 152) and G (10, –157)

     
    d) F (62, 50) and E (62, –26)

     

    		 e) D (–3, 5) and C (–3, –9)

     

    		 f) B (–6, 6) and A (–6, –2)

     

  3. Find the length of the line segment formed by each pair of points. Write your answers as a radical in simplest form and as a decimal rounded to the nearest hundredth.

    a) A (2, 3) and B (5, 6)

     

     

     

    		 b) C (–4, –1) and D (1, 2)

     

     

     

    		 c) E (3, –7) and F (8, –2)

     

     

     
    d) G (–6, 4) and H (–2, 1)

     

     

     

    		 e) I (7, 5) and J (2, 1)

     

     

     

    		 f) K (4, 9) and L (–2, 4)

     

     

     

  4. For each pair of points, draw the line segment on the grid and find its length.

    Blank coordinate plane. Both axes have a minimum of -12 and maximum of 12 and a scale of 2. a) M (–3, 5) and N (2, 5)

    b) P (–4, –2) and Q (1, –2)

    c) X (6, 9) and Y (2, 9)

    d) L (4, 7) and K (4, –3)

    e) J (–2, 8) and I (–2, 5)

    f) H (10, 6) and G (10, –2)

  5. A rectangle has vertices A (–45, –2), B (–12, –2), C (–12, –25), and D (–45, –25). Find the perimeter and area of the rectangle.

     

     

     

     

  6. An isosceles triangle has vertices N (–4, –6), M (–4, 6), and O (8, 0). Draw the triangle on the grid below. Find the area and perimeter of the shape. Round to two decimal places where necessary.

    Blank coordinate plane. Both axes have a minimum of -12 and maximum of 12 and a scale of 2.

  7. Given that each line segment is either a vertical or horizontal line, find x or y.

    a) A (x, 3) and B (–5, 9)

     

     

     

    		 b) C (–10, 9) and D (15, y)

     

     

     

    		 c) E (x, 3) and F (3, –9)

     

     

     

  8. Connect the points on the following grid to form a pentagon. To draw the pentagon, start at point (–3, 1) and draw clockwise. Find the perimeter of the shape. Round your answer to the nearest tenth.

    Graph with points (-3,1), (-1,4), (0,-2), (2,4), and (4,-3).

Answers

1.

a) 8 b) 6 c) 7 d) 67 e) 10 f) 105

2.

a) 10 b) 3 c) 309 d) 76 e) 14 f) 8

3.

a) 3\sqrt{2}
4.24		 b) \sqrt{34}
5.83		 c) 5\sqrt{2}
7.07		 d) 5 e) \sqrt{41}
6.40		 f) \sqrt{61}
7.81

4.

a) 5 b) 5 c) 4 d) 10 e) 3 f) 8
5. P = 112 units and A = 759 units2 6. P = 38.83 units and A = 72 units2

7.

a) x = –5 b) y = 9 c) x = 3

8. P = 22.2 units

 

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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Math 10C Workbook Copyright © 2026 by Leanne Thompson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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