 # Find the Midpoint

In this worksheet, students will find the coordinates of midpoints of line segments with coordinates involving both numbers and algebra, or use knowledge of midpoints to find missing end or starting points. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra, Graphs of Equations and Functions

Curriculum subtopic:   Graphs, Straight Line Graphs

Difficulty level:   ### QUESTION 1 of 10

To find the middle point between two numbers, we can add them together and then divide by 2.

e.g. The middle point between 4 and 7 is 4 + 7 = 11, 11 ÷ 2 = 5.5

We can use the same method to find the midpoint (the middle) of a line segment.

A line segment is simply a part of a line between two points.

A line segment has a beginning and an end.

e.g. Let's say we have a line segment between (1, 2) and (5, 6). What is the midpoint of this line segment?

We need to find the middle between the x-coordinates and the y-coordinates.

Simply add the x-coordinates and divide by 2, then add the y-coordinates and divide by 2.

x-coordinates:

1 + 5 = 6

6 ÷ 2 = 3

y-coordinates:

2 + 6 = 8

8 ÷ 2 = 4

So the coordinates of the midpoint of this line segment are (3, 4).

In this activity, we will find the midpoints of line segments with coordinates involving both numbers and algebra, or use our knowledge of midpoints to find missing end or starting points.

Find the midpoint of the line segment between (1, 8) and (9, 2).

(10, 10)

(1.5 , 8.3)

(5, 5)

(6, 7)

Find the midpoint of the line segment between (6, 3) and (-2, 4).

Do not use any spaces between your numbers and the brackets or comma or you may be marked incorrectly.

Find the midpoint of the line segment between (0, -3) and (7, -1).

(3.5, -1)

(7, -4)

(3.5, -2)

(3.5, 2)

Match each pair of points below with their midpoints.

## Column B

(-2, 5) and (4, -1)
(-4.5, -2,5)
(-4, -2) and (2, 4)
(1, 2)
(-2, 4) and (4, 2)
(1, 3)
(-7 -4) and (-2, -1)
(-1, 1)

Find the midpoint of the line segment between (-7, -2) and (3, -5).

Do not use any spaces between your numbers and the brackets or comma or you may be marked incorrectly.

A line segment has a starting point (4, 5).

Its midpoint is (6, 8).

What are the coordinates of the end point of this line segment?

(5, 6.5)

(8, 11)

(6.5, 5)

(10, 13)

True or false?

If the starting point of a line segment is (3, 11) and the midpoint is (7, 17), its end point must be (5, 14).

True

False

The midpoint of a line segment AB is (-3, 3).

Point A has coordinates (-6, 8).

What are the coordinates of Point B?

Do not use any spaces between your numbers and the brackets or comma or you may be marked incorrectly.

Find the coordinates of a midpoint of a line segment AB, where A and B have coordinates:

A = (2p, q)

B = (6p, 7q)

(10p, 13q)

(8p, 8q)

(4p, 4q)

(-2p, -5q)

Find the coordinates of a midpoint of a line segment AB, where A and B have coordinates:

A = (8p, 2q)

B = (2p, 14q)

Do not use any spaces between your numbers and the brackets or comma or you may be marked incorrectly.

• Question 1

Find the midpoint of the line segment between (1, 8) and (9, 2).

(5, 5)
EDDIE SAYS
Let's follow the method from our Introduction. Remember, add the corresponding coordinates and divide by 2. x-coordinates: 1 + 9 = 10 10 ÷ 2 = 5 y-coordinates: 8 + 2 = 10 10 ÷ 2 = 5 Then we need to put those coordinates back together to describe our midpoint: (5, 5) Does that make sense? Review the information in the Introduction now to make sure you are totally confident with this process before attempting the rest of this activity.
• Question 2

Find the midpoint of the line segment between (6, 3) and (-2, 4).

Do not use any spaces between your numbers and the brackets or comma or you may be marked incorrectly.

(2,3.5)
(2, 3.5)
EDDIE SAYS
Let's add the corresponding coordinates and divide by 2. x-coordinates: 6 + (-2) = 4 4 ÷ 2 = 2 y-coordinates: 3 + 4 = 7 7 ÷ 2 = 3.5 Then we need to put those coordinates back together to describe our midpoint: (2, 3.5) Be careful with those symbols when we are working with positive and negative coordinates.
• Question 3

Find the midpoint of the line segment between (0, -3) and (7, -1).

(3.5, -2)
EDDIE SAYS
Follow the same method as we did before, but be sure to take care with those negative numbers! 0 + 7 = 7 7 ÷ 2 = 3.5 -3 + (-1) = -4 -4 ÷ 2 = -2 So the midpoint is (3.5, -2).
• Question 4

Match each pair of points below with their midpoints.

## Column B

(-2, 5) and (4, -1)
(1, 2)
(-4, -2) and (2, 4)
(-1, 1)
(-2, 4) and (4, 2)
(1, 3)
(-7 -4) and (-2, -1)
(-4.5, -2,5)
EDDIE SAYS
Here we need to follow the same method for each pair of points and then find the midpoint which matches. Let's look at one example together: (-2, 5) and (4, -1). x-coordinates: (-2) + 4 = 2 2 ÷ 2 = 1 y-coordinates: 5 + (-1) = 4 4 ÷ 2 = 2 So our midpoint coordinates here are: (1, 2) Can you use this example and process to find the remaining three matches independently?
• Question 5

Find the midpoint of the line segment between (-7, -2) and (3, -5).

Do not use any spaces between your numbers and the brackets or comma or you may be marked incorrectly.

(-2,-3.5)
(-2, -3.5)
EDDIE SAYS
There's quite a few negative numbers here, so we need to be careful when we add them. -7 + 3 = -4 4 ÷ 2 = -2 -2 + -5 = -7 -7 ÷ 2 = -3.5 The midpoint is (-2, -3.5).
• Question 6

A line segment has a starting point (4, 5).

Its midpoint is (6, 8).

What are the coordinates of the end point of this line segment?

(8, 11)
EDDIE SAYS
Here we need to work backwards, as we have one of the starting point and the midpoint, but need to find the end point. The midpoint is (6, 8). This means that after adding the coordinates in the process we have used before, we will reach: (12, 16) (Because then we would divide by 2 to get (6, 8)). So our x-coordinates must add to 12. 12 - 4 = 8, so the end point has an x-coordinate of 8. Our y-coordinates must add to 16. 16 - 5 = 11, so the end point has a y-coordinate of 11. So the coordinates of the endpoint are: (8, 11)
• Question 7

True or false?

If the starting point of a line segment is (3, 11) and the midpoint is (7, 17), its end point must be (5, 14).

False
EDDIE SAYS
This statement is false. If the starting point is (3, 11) and the midpoint is (7, 17), then the coordinates of the start and the end points must add to (14, 34) (double the coordinates of the midpoint). 14 - 3 = 11 (x-coordinate) 34 - 11 = 23 (y-coordinate) So the coordinates of the end point are (11, 23) not (5, 14) as this statement claims.
• Question 8

The midpoint of a line segment AB is (-3, 3).

Point A has coordinates (-6, 8).

What are the coordinates of Point B?

Do not use any spaces between your numbers and the brackets or comma or you may be marked incorrectly.

(0,-2)
(0, -2)
EDDIE SAYS
The question describes the line segment in the terms 'AB', but this simply means that A is the starting point and B is the end point. The coordinates of the midpoint are (-3, 3) so the coordinates of A and B must add to double that, which is (-6, 6). -6 - (-6) = 6 + 6 = 0 So the x-coordinate of B is 0. 6 - 8 = -3 So the y-coordinate of B is -3. In summary, the coordinates of point B are (0, -3).
• Question 9

Find the coordinates of a midpoint of a line segment AB, where A and B have coordinates:

A = (2p, q)

B = (6p, 7q)

(4p, 4q)
EDDIE SAYS
Don't be put off by this question! The coordinates are given in an algebraic form, but the same method can be used to find the midpoint of this line segment: Add the corresponding coordinates and divide by 2. 2p + 6p = 8p 8p ÷ 2 = 4p (x-coordinate) q + 7q = 8q 8q ÷ 2 = 4q (y-coordinate) So the coordinates of the midpoint are (4p, 4q).
• Question 10

Find the coordinates of a midpoint of a line segment AB, where A and B have coordinates:

A = (8p, 2q)

B = (2p, 14q)

Do not use any spaces between your numbers and the brackets or comma or you may be marked incorrectly.

(5p,8q)
(5p, 8q)
EDDIE SAYS
Here's another questions involving algebra. Have a look below to see the method to find the correct midpoint of the line segment AB. 8p + 2p = 10p 10p ÷ 2 = 5p (x-coordinate) 2q + 14q = 16q 16q ÷ 2 = 8q (y-coordinate) So the coordinates of the midpoint are (5p, 8q). Congratulations on completing this activity. Why not try another activity focusing on graphs in algebra, as you are on a roll!
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