P(−8, 4) lies on the segment AB, with A(1, 1) and B(−11, 5). AP : PB = □ : □ (in lowest terms) Fill in each blank

3, 1. Look at x: AP goes −8 − (1) = −9 and PB −11 − (−8) = −3, so AP : PB = −9 : −3 = 3 : 1.

A(6, −5), B(−6, 7). The point P on AB with AP : PB = 3 : 1 is (3, −2). True or false?

False. P = ((1 × (6) + 3 × (−6))/4, (1 × (−5) + 3 × (7))/4) = (−3, 4); P is nearer the end with the smaller share.

P lies on the segment AB with AP : PB = 1 : 1, A(3, −6) and P(0, −7). B = (□, □). Fill in each blank

−3, −8. From A to P is 1 parts, each changing x by −3 and y by −1; 1 more parts from P reach B: (0 + (−3), −7 + (−1)) = (−3, −8).

Coordinate geometry · Ages 14–15

The section formula (internal division)

s times x₁ and r times x₂, over r + s.

How to do it

Method

s times x₁ and r times x₂, over r + s

AP : PB = 1 : 2: P = (2 × 1 + 1 × 73, 2 × 2 + 1 × 113) = (3, 5)

Another way

Walk r/(r + s) of the way

1 + 6 × 13 = 3, 2 + 9 × 13 = 5

In full

P on the segment AB with AP : PB = r : s divides AB internally. The formula: P = (s x₁ + r x₂r + s, s y₁ + r y₂r + s); note x₁ is multiplied by s and x₂ by r: the end P is nearer weighs more. A(1, 2), B(7, 11), AP : PB = 1 : 2: P = (2 × 1 + 1 × 73, 2 × 2 + 1 × 113) = (3, 5). Another way: walk it. AP is rr + s of AB, so from A, x goes 13 of (7 − 1) = 2 and y 13 of (11 − 2) = 3, to (1 + 2, 2 + 3) = (3, 5), the same answer. With r : s = 1 : 1, P is the mid-point. For the ratio from P, look at the x-coordinates: AP : PB = (x − x₁) : (x₂ − x).

Worked examples

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

Example 1
P(−8, 4) lies on the segment AB, with A(1, 1) and B(−11, 5). AP : PB = : (in lowest terms)
Fill in each blank
Show the answer and working
Answer: 3, 1
Look at x: AP goes −8 − (1) = −9 and PB −11 − (−8) = −3, so AP : PB = −9 : −3 = 3 : 1.
Example 2
A(6, −5), B(−6, 7). The point P on AB with AP : PB = 3 : 1 is (3, −2).
True or false?
Show the answer and working
Answer: False
P = (1 × (6) + 3 × (−6)4, 1 × (−5) + 3 × (7)4) = (−3, 4); P is nearer the end with the smaller share.
Example 3
P lies on the segment AB with AP : PB = 1 : 1, A(3, −6) and P(0, −7). B = (, ).
Fill in each blank
Show the answer and working
Answer: −3, −8
From A to P is 1 parts, each changing x by −3 and y by −1; 1 more parts from P reach B: (0 + (−3), −7 + (−1)) = (−3, −8).

Practice questions to try

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

1.P lies on the segment AB, with A(−6, 5), B(−18, 17) and AP : PB = 3 : 1. P = (, ).
Fill in each blank
Answer:
2.A(−2, 5), B(−14, 17), and P on AB with AP : PB = 3 : 1. From A to B, x changes by and y by ; AP is 34 of AB, so P = (, ).
Fill in each blank
Answer:
3.A(5, 5), B(20, −10). The point P on AB with AP : PB = 2 : 3 is (14, −4).
True or false?
Answer:
4.A(0, −3), B(−8, −4), and P on AB with AP : PB = 4 : 3. P's x-coordinate = (as a fraction)
Answer in simplest form
Answer:
5.P lies on the segment AB with AP : PB = 1 : 2, A(−2, 4) and P(1, 5). B = (, ).
Fill in each blank
Answer:

All the answers

1. −15, 14
2. −12, 12, −11, 14
3. False
4. −327
5. 7, 7

Free worksheet: The section formula (internal division)Working 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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