The segment AB has one end at A(1, 3) and its mid-point at M(−2, −5). The other end is B = (□, □). Fill in each blank

−5, −13. (1 + x)/2 = −2, so x = 2 × (−2) − (1) = −4 − (1) = −5; likewise y = −10 − (3) = −13. As far again from M as from A to M.

M(−1, −1) is the mid-point of A(2, 4) and B(−4, −6). True or false?

True. The mid-point is ((2 + (−4))/2, (4 + (−6))/2) = (−1, −1).

Going from A(−5, 2) to B(3, 4), x changes by □ and y by □. The mid-point is half the way, so it is (□, □). Fill in each blank

8, 2, −1, 3. x changes by 8, half is 4: −5 + (4) = −1; y by 2, half 1: 2 + (1) = 3. The same as the averages, ((−5 + (3))/2, …).

Coordinate geometry · Ages 14–15

The mid-point formula

The mid-point is the average of the ends.

How to do it

Method

The mid-point is the average of the ends

A(2, 7), B(8, −1): M = (5, 3)

Another way

Half the way from A

2 + 6 ÷ 2 = 5, 7 − 8 ÷ 2 = 3

In full

The mid-point M of AB sits exactly halfway, and its coordinates are the averages of the ends': M = (x₁ + x₂2, y₁ + y₂2). A(2, 7), B(8, −1): M = (2 + 82, 7 + (−1)2) = (5, 3). Another way to see it: from A to B, x goes up 8 − 2 = 6 and y down 8; the mid-point is half the way, so from A go 3 right and 4 down, to (5, 3), the same answer. Given the mid-point and one end, turn the formula round: x₂ = 2 × (the mid-point's x) − x₁; or walk it: as far again from M as from A to M. M(5, 3), A(2, 7): B = (2 × 5 − 2, 2 × 3 − 7) = (8, −1).

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
The segment AB has one end at A(1, 3) and its mid-point at M(−2, −5). The other end is B = (, ).
Fill in each blank
Show the answer and working
Answer: −5, −13
1 + x2 = −2, so x = 2 × (−2) − (1) = −4 − (1) = −5; likewise y = −10 − (3) = −13. As far again from M as from A to M.
Example 2
M(−1, −1) is the mid-point of A(2, 4) and B(−4, −6).
True or false?
Show the answer and working
Answer: True
The mid-point is (2 + (−4)2, 4 + (−6)2) = (−1, −1).
Example 3
Going from A(−5, 2) to B(3, 4), x changes by and y by . The mid-point is half the way, so it is (, ).
Fill in each blank
Show the answer and working
Answer: 8, 2, −1, 3
x changes by 8, half is 4: −5 + (4) = −1; y by 2, half 1: 2 + (1) = 3. The same as the averages, (−5 + (3)2, …).

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.The mid-point of A(6, −4) and B(−2, 6) is M = (, ).
Fill in each blank
Answer:
2.The segment AB runs from A(5, 3) to B(−3, 5). Which point is its mid-point?
- A. (4, 1)
- B. (−4, 1)
- C. (2, 8)
- D. (1, 4)
Choose the right answer
Answer:
3.Read A and B off the picture. The mid-point of AB is (, ).
Fill in each blank
Answer:
4.Going from A(0, 3) to B(−2, −5), x changes by and y by . The mid-point is half the way, so it is (, ).
Fill in each blank
Answer:
5.The mid-point of A(−8, 2) and B(−5, −2) has x-coordinate (as a fraction)
Answer in simplest form
Answer:

All the answers

1. 2, 1
2. D. (1, 4)
3. −5, 2
4. −2, −8, −1, −1
5. −132

Free worksheet: The mid-point formulaWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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