Coordinates and trigonometry · Ages 14–15
Coordinate geometry worksheet
Covers: The distance formula, The mid-point formula, The section formula (internal division), Slope, Using slopes, Slopes of parallel and perpendicular lines, Proofs with coordinates, External division
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Coordinate geometry worksheet
Coordinates and trigonometry · Ages 14–15
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.Find the distance between A(−8, −6) and B(−8, −7): AB =
Answer:
2.A(0, −3), B(−1, 4). AB =
Answer:
3.A(−2, −1), B(−6, −3), and P on AB with AP : PB = 1 : 1. From A to B, x changes by and y by ; AP is 12 of AB, so P = (, ).
Fill in each blank
Answer:
4.A line through A(1, 1) and B(2, k) has slope −21. k =
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5.P lies on BA extended (A between P and B), with A(6, 2), B(8, 5) and AP : PB = 1 : 2. P = (, ).
Fill in each blank
Answer:
Part B: Choose, or true or false (2 marks each)
6.The segment AB runs from A(1, 6) to B(−3, 4). Which point is its mid-point?
- A. (−2, −1)
- B. (−1, 5)
- C. (5, −1)
- D. (−2, 10)
Choose the right answer
Answer:
7.M(1, 0) is the mid-point of A(6, −5) and B(−6, 5).
True or false?
Answer:
8.A(−6, 1), B(3, 10). P is on AB with AP : PB = 1 : 2. Which point is P?
- A. (−9, 12)
- B. (−32, 112)
- C. (0, 7)
- D. (−3, 4)
Choose the right answer
Answer:
9.What is the slope of the line through A(−4, −5) and B(−6, −5)?
- A. Undefined
- B. Positive
- C. Negative
- D. 0
Choose the right answer
Answer:
10.The line through A(−1, −4) and B(1, 6) has slope 51.
True or false?
Answer:
11.Lines of slopes −23 and −32 are perpendicular.
True or false?
Answer:
12.A(1, 4), B(0, 5), C(2, 7). ∠ABC = 90°.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.Home is at the origin O(0, 0). Four corner shops are at P(−4, 4), Q(−6, 6), R(−3, −6) and S(−1, −3). Which is nearest home in a straight line?
- A. R
- B. P
- C. Q
- D. S
Choose the right answer
Working:
Answer:
14.Two friends live at A(0, −5) and B(−4, −7) and agree to meet at the mid-point of the line between their homes. They meet at (, ).
Fill in each blank
Working:
Answer:
15.A straight road joins town A(−2, −1) to town B(8, −16). A petrol station P is to stand on it with AP : PB = 3 : 2. P = (, ).
Fill in each blank
Working:
Answer:
16.A straight sloping path has slope 111. Walking up it, you move 44 m across. How many metres do you rise?
Working:
Answer:
17.On a plan, corridor AB runs from A(1, −4) to B(0, −5) and corridor CD from C(2, 1) to D(4, 0). The corridors are perpendicular.
True or false?
Working:
Answer:
18.A triangular flower bed has corners at A(−2, −1), B(−5, 0) and C(−4, 3). Prove ∠ABC = 90° from its sides' lengths: put the steps in order.
- A. ∠ABC = 90° (converse of Pythagoras' theorem)
- B. AB2 + BC2 = 10 + 10
- C. = 20
- D. = AC2
Write the letters in the right order
Working:
Answer:
Show the answers and working
Coordinate geometry worksheet: Answers and working
Part A: Working out and filling in
1. 1
The points share a coordinate, so they are on one horizontal or vertical line: the distance is how far the other coordinate differs, 1.
2. 50
AB = (−1 − (0))2 + (4 − (−3))2 = 1 + 49 = 50. Bracket a negative coordinate as you subtract.
3. −4, −2, −4, −2
x changes by −4, and 12 of it is −2: −2 + (−2) = −4; y by −2, 12 of it −1: −1 + (−1) = −2.
4. −1
k − (1)2 − (1) = −21, so k − (1) = −21 × 1 = −2 and k = −1.
5. 4, −1
PB is 2 parts and AP 1, so AB is 1 parts, each changing x by 2 and y by 3; back 1 parts from A: (4, −1). The formula (1 × (8) − 2 × (6)1 − 2, …) agrees.
Part B: Choose, or true or false
6. B. (−1, 5)
M = (1 + (−3)2, 6 + (4)2) = (−1, 5).
7. False
The mid-point is (6 + (−6)2, −5 + (5)2) = (0, 0).
8. D. (−3, 4)
P = (2 × (−6) + 1 × (3)3, 2 × (1) + 1 × (10)3) = (−3, 4).
9. D. 0
Look at −5 − (−5)−6 − (−4): same signs, positive; opposite, negative; 0 on top, a slope of 0; 0 below, no slope is defined.
10. True
Slope = 6 − (−4)1 − (−1) = 102 = 51. Mind the sign: subtract in the same order on top and below.
11. False
Perpendicular means the slopes multiply to −1: work out −23 × −32 and see whether it is −1.
12. True
Find the slopes of AB and BC: if they multiply to −1 they are perpendicular and ∠ABC = 90°; or see whether AB2 + BC2 = AC2.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. D. S
Comparing the squared distances is enough: P 32, Q 72, R 45, S 10; the smallest is nearest.
14. −2, −6
The mid-point is (0 + (−4)2, −5 + (−7)2) = (−2, −6).
15. 4, −10
P = (2 × (−2) + 3 × (8)5, 2 × (−1) + 3 × (−16)5) = (4, −10).
16. 4 m
Rise ÷ 44 = 111, so the rise = 111 × 44 = 4 m.
17. False
Find both corridors' slopes and see whether they multiply to −1 (if one is upright, see whether the other is level).
18. B → C → D → A
AB2 = 10 and BC2 = 10 add to 20, which is AC2; the sides fit Pythagoras, so by the converse ∠ABC = 90°.
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