Coordinates and trigonometry · Ages 14–15

The section formula (internal division) worksheet

Part of: Coordinate geometry

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Coordinates and trigonometry · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.P lies on the segment AB, with A(1, 6), B(−4, 1) and AP : PB = 3 : 2. P = (, ).
Fill in each blank
Answer:
2.A(2, −2), B(12, −7), and P on AB with AP : PB = 2 : 3. From A to B, x changes by and y by ; AP is 25 of AB, so P = (, ).
Fill in each blank
Answer:
3.P(−8, −3) lies on the segment AB, with A(−6, −5) and B(−9, −2). AP : PB = : (in lowest terms)
Fill in each blank
Answer:
4.P(7, −5) lies on the segment AB, with A(6, −6) and B(9, −3). AP : PB = : (in lowest terms)
Fill in each blank
Answer:
5.P lies on the segment AB with AP : PB = 3 : 1, A(1, −6) and P(10, −3). B = (, ).
Fill in each blank
Answer:
6.A(−7, −8), B(6, −6), and P on AB with AP : PB = 3 : 2. P's x-coordinate = (as a fraction)
Answer in simplest form
Answer:
7.P lies on the segment AB with AP : PB = 2 : 1, A(5, −3) and P(7, −9). B = (, ).
Fill in each blank
Answer:
8.A(3, −8), B(1, 5), and P on AB with AP : PB = 4 : 3. P's x-coordinate = (as a fraction)
Answer in simplest form
Answer:

9.A(1, −2), B(−2, 4). The point P on AB with AP : PB = 2 : 1 is (−1, 2).
True or false?
Answer:
10.A(2, −2), B(6, −10). P is on AB with AP : PB = 1 : 3. Which point is P?
- A. (5, −8)
- B. (41, −61)
- C. (3, −4)
- D. (12, −16)
Choose the right answer
Answer:
11.A(−1, −2), B(−13, 6). P is on AB with AP : PB = 1 : 3. Which point is P?
- A. (−10, 4)
- B. (−4, 0)
- C. (−71, 21)
- D. (−16, 0)
Choose the right answer
Answer:
12.A(4, −1), B(7, 5). The point P on AB with AP : PB = 1 : 2 is (6, 3).
True or false?
Answer:

13.A straight road joins town A(−6, −4) to town B(9, −9). A petrol station P is to stand on it with AP : PB = 3 : 2. P = (, ).
Fill in each blank
Working:
Answer:
14.3 trees are planted on the straight line between A(−2, −1) and B(10, 11), cutting AB into 4 equal parts (none at the ends). Counting from A, tree number 3 is at (, ).
Fill in each blank
Working:
Answer:
15.A cable car runs in a straight line from station A(−3, 6) to station B(−13, −9) and has gone 35 of the way. From A to B, x changes by and y by ; the car is now at (, ).
Fill in each blank
Working:
Answer:
16.A runner goes in a straight line from A(−4, −5) towards B(17, 16) and is now at P(5, 4). The distance run and the distance left are in the ratio : (in lowest terms).
Fill in each blank
Working:
Answer:
17.A runner goes in a straight line from A(−3, 2) towards B(2, 17) and is now at P(−1, 8). The distance run and the distance left are in the ratio : (in lowest terms).
Fill in each blank
Working:
Answer:
18.A straight road joins town A(−6, 2) to town B(−21, −13). A petrol station P is to stand on it with AP : PB = 2 : 3. P = (, ).
Fill in each blank
Working:
Answer:

Show the answers and working

The section formula (internal division) worksheet: Answers and working

1. −2, 3
P = (2 × (1) + 3 × (−4)5, 2 × (6) + 3 × (1)5) = (−2, 3). x₁ takes s, x₂ takes r.
2. 10, −5, 6, −4
x changes by 10, and 25 of it is 4: 2 + (4) = 6; y by −5, 25 of it −2: −2 + (−2) = −4.
3. 2, 1
Look at x: AP goes −8 − (−6) = −2 and PB −9 − (−8) = −1, so AP : PB = −2 : −1 = 2 : 1.
4. 1, 2
Look at x: AP goes 7 − (6) = 1 and PB 9 − (7) = 2, so AP : PB = 1 : 2 = 1 : 2.
5. 13, −2
From A to P is 3 parts, each changing x by 3 and y by 1; 1 more parts from P reach B: (10 + (3), −3 + (1)) = (13, −2).
6. 45
x = 2 × (−7) + 3 × (6)5 = 45 = 45.
7. 8, −12
From A to P is 2 parts, each changing x by 1 and y by −3; 1 more parts from P reach B: (7 + (1), −9 + (−3)) = (8, −12).
8. 137
x = 3 × (3) + 4 × (1)7 = 137 = 137.

9. True
P = (1 × (1) + 2 × (−2)3, 1 × (−2) + 2 × (4)3) = (−1, 2); P is nearer the end with the smaller share.
10. C. (3, −4)
P = (3 × (2) + 1 × (6)4, 3 × (−2) + 1 × (−10)4) = (3, −4).
11. B. (−4, 0)
P = (3 × (−1) + 1 × (−13)4, 3 × (−2) + 1 × (6)4) = (−4, 0).
12. False
P = (2 × (4) + 1 × (7)3, 2 × (−1) + 1 × (5)3) = (5, 1); P is nearer the end with the smaller share.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 3, −7
P = (2 × (−6) + 3 × (9)5, 2 × (−4) + 3 × (−9)5) = (3, −7).
14. 7, 8
Each part changes x by 3 and y by 3. Tree 3 is 34 of the way, AP : PB = 3 : (4 − 3): (−2 + 3 × (3), −1 + 3 × (3)) = (7, 8).
15. −10, −15, −9, −3
x changes by −10, 35 of it −6; y by −15, 35 of it −9. The car is at (−3 + (−6), 6 + (−9)) = (−9, −3).
16. 3, 4
Look at x: run 5 − (−4) = 9, left 17 − (5) = 12, a ratio of 9 : 12 = 3 : 4.
17. 2, 3
Look at x: run −1 − (−3) = 2, left 2 − (−1) = 3, a ratio of 2 : 3 = 2 : 3.
18. −12, −4
P = (3 × (−6) + 2 × (−21)5, 3 × (2) + 2 × (−13)5) = (−12, −4).

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