Coordinates and trigonometry · Ages 14–15

External division worksheet

Part of: Coordinate geometry

In the print window, choose “Save as PDF” to download it. The answers print on the last page.

Coordinates and trigonometry · Ages 14–15

Numsprite
numsprite.com

Name:Date:Score:／48

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

1.P lies on AB extended, with A(3, −6), B(1, −7) and AP : PB = 5 : 4. P = (, ).
Fill in each blank
Answer:
2.P lies on AB extended, with A(−3, −6), B(0, −4) and AP : PB = 2 : 1. P = (, ).
Fill in each blank
Answer:
3.P lies on BA extended (A between P and B), with A(1, −4), B(3, −2) and AP : PB = 3 : 4. P = (, ).
Fill in each blank
Answer:
4.P lies on BA extended (A between P and B), with A(−1, −2), B(−2, 1) and AP : PB = 2 : 3. P = (, ).
Fill in each blank
Answer:
5.P lies on AB extended (B between A and P), with AP : PB = 2 : 1, A(−1, 6) and P(−5, 8). B = (, ).
Fill in each blank
Answer:
6.A(−8, −8), B(−3, 5). P is on the line AB but not between A and B, with AP : PB = 5 : 2. P's x-coordinate = (as a fraction)
Answer in simplest form
Answer:
7.P lies on AB extended (B between A and P), with AP : PB = 5 : 3, A(3, 0) and P(13, 15). B = (, ).
Fill in each blank
Answer:
8.A(4, −2), B(−6, 8). P is on the line AB but not between A and B, with AP : PB = 5 : 1. P's x-coordinate = (as a fraction)
Answer in simplest form
Answer:

9.A(−3, 0), B(−4, −6). P is on the line AB but not between A and B, with AP : PB = 3 : 5. B lies between A and P.
True or false?
Answer:
10.A(−4, 5), B(4, 0). P is on the line AB but not between A and B, with AP : PB = 1 : 2. B lies between A and P.
True or false?
Answer:
11.A(−3, −6), B(6, −9). P lies on AB extended with AP : PB = 4 : 1. Which point is P?
- A. (215, −425)
- B. (15, −12)
- C. (9, −10)
- D. (−61, −51)
Choose the right answer
Answer:
12.A(−6, 1), B(3, 10). P lies on AB extended with AP : PB = 4 : 1. Which point is P?
- A. (−91, −21)
- B. (65, 415)
- C. (12, 19)
- D. (6, 13)
Choose the right answer
Answer:

13.A straight runway runs from A(−2, 4) through B(1, 7) on to P, with AP : PB = 4 : 1. P = (, ).
Fill in each blank
Working:
Answer:
14.A straight water pipe is laid from A(3, −1) through B(2, −3) on to P, with AP : PB = 3 : 2. Taking AP as 3 parts and PB as 2, AB is parts and BP parts; P = (, ).
Fill in each blank
Working:
Answer:
15.A laser beam goes from A(1, 3) towards B(−2, 6), passes B and carries straight on to P on a wall, with AP : PB = 4 : 3. P's y-coordinate =
Working:
Answer:
16.A route goes in a straight line from the start A(6, −6) to the finish P(9, −3), passing the supply point B on the way, with AP : PB = 3 : 2. B = (, ).
Fill in each blank
Working:
Answer:
17.A straight runway runs from A(−1, −1) through B(−3, −5) on to P, with AP : PB = 3 : 1. P = (, ).
Fill in each blank
Working:
Answer:
18.A laser beam goes from A(−3, −4) towards B(−6, −2), passes B and carries straight on to P on a wall, with AP : PB = 3 : 2. P's y-coordinate =
Working:
Answer:

Show the answers and working

External division worksheet: Answers and working

1. −7, −11
External division: P = (5 × (1) − 4 × (3)1, 5 × (−7) − 4 × (−6)1) = (−71, −111) = (−7, −11).
2. 3, −2
External division: P = (2 × (0) − 1 × (−3)1, 2 × (−4) − 1 × (−6)1) = (31, −21) = (3, −2).
3. −5, −10
PB is 4 parts and AP 3, so AB is 1 parts, each changing x by 2 and y by 2; back 3 parts from A: (−5, −10). The formula (3 × (3) − 4 × (1)3 − 4, …) agrees.
4. 1, −8
PB is 3 parts and AP 2, so AB is 1 parts, each changing x by −1 and y by 3; back 2 parts from A: (1, −8). The formula (2 × (−2) − 3 × (−1)2 − 3, …) agrees.
5. −3, 7
AP is 2 parts and PB 1, so AB is 1 parts, each changing x by −2 and y by 1: B = (−1 + 1 × (−2), 6 + 1 × (1)) = (−3, 7).
6. 13
x = 5 × (−3) − 2 × (−8)5 − 2 = 13 = 13.
7. 7, 6
AP is 5 parts and PB 3, so AB is 2 parts, each changing x by 2 and y by 3: B = (3 + 2 × (2), 0 + 2 × (3)) = (7, 6).
8. −172
x = 5 × (−6) − 1 × (4)5 − 1 = −344 = −172.

9. False
When AP is longer than PB (3 > 5), P is beyond B, so B is between A and P; when AP is shorter, P is beyond A.
10. False
When AP is longer than PB (1 > 2), P is beyond B, so B is between A and P; when AP is shorter, P is beyond A.
11. C. (9, −10)
P = (4 × (6) − 1 × (−3)3, 4 × (−9) − 1 × (−6)3) = (9, −10).
12. D. (6, 13)
P = (4 × (3) − 1 × (−6)3, 4 × (10) − 1 × (1)3) = (6, 13).

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

13. 2, 8
P = (4 × (1) − 1 × (−2)3, 4 × (7) − 1 × (4)3) = (2, 8).
14. 1, 2, 0, −7
AB = 3 − 2 = 1 parts, each changing x by −1 and y by −2; 2 more parts from B: (2 + (−2), −3 + (−4)) = (0, −7).
15. 15
y = 4 × (6) − 3 × (3)4 − 3 = 15.
16. 7, −5
AP is 3 parts, PB 2 and AB 1: B is 13 of the way along AP, at (6 + 1 × (1), −6 + 1 × (1)) = (7, −5).
17. −4, −7
P = (3 × (−3) − 1 × (−1)2, 3 × (−5) − 1 × (−1)2) = (−4, −7).
18. 2
y = 3 × (−2) − 2 × (−4)3 − 2 = 2.

Numsprite · numsprite.com/en/worksheets/coordinate-geometry/external-point-of-division