Coordinates and trigonometry · Ages 14–15

Proofs with coordinates 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.A(−3, −3), B(−2, −2), C(0, −4). Prove ∠ABC = 90° by the lengths: put the steps in order.
- A. AB² + BC² = 2 + 8
- B. = 10
- C. ∠ABC = 90° (converse of Pythagoras' theorem)
- D. = AC²
Write the letters in the right order
Answer:
2.A(4, −4), B(6, −3), C(7, −5). AB² = , BC² = , AC² = ; since AB² + BC² = AC², ∠ABC = 90°.
Fill in each blank
Answer:
3.A(−8, 4), B(−6, 1), C(−12, −3). Prove ∠ABC = 90° by the lengths: put the steps in order.
- A. = 65
- B. = AC²
- C. ∠ABC = 90° (converse of Pythagoras' theorem)
- D. AB² + BC² = 13 + 52
Write the letters in the right order
Answer:
4.A(−1, −5), B(−4, −6), C(−6, 0). AB² = , BC² = , AC² = ; since AB² + BC² = AC², ∠ABC = 90°.
Fill in each blank
Answer:
5.ABCD is a parallelogram with A(6, −3), B(−2, 3), C(6, 2). Its diagonals bisect each other, so AC's mid-point is BD's, and D = (, ).
Fill in each blank
Answer:

6.A(−4, 4), B(3, 1), C(2, 0), D(−6, −3). To prove AB ∥ CD with coordinates, which should be shown?
- A. AB's slope = CD's slope
- B. AB's mid-point = CD's mid-point
- C. AB = CD
- D. AB's slope × CD's slope = −1
Choose the right answer
Answer:
7.A(1, −4), B(0, −5), C(−1, −4). ∠ABC = 90°.
True or false?
Answer:
8.A(5, 8), B(2, 6), C(0, 9). ∠ABC = 90°.
True or false?
Answer:
9.A(1, −2), B(−5, −6), C(2, −3), D(−5, 0). To prove AB ∥ CD with coordinates, which should be shown?
- A. AB's mid-point = CD's mid-point
- B. AB's slope × CD's slope = −1
- C. AB's slope = CD's slope
- D. AB = CD
Choose the right answer
Answer:
10.A(−4, −4), B(−1, −6), C(−5, 6), D(−8, 8). ABCD is a parallelogram.
True or false?
Answer:
11.A(6, 6), B(5, 1), C(−3, −1), D(−2, 3). ABCD is a parallelogram.
True or false?
Answer:
12.What kind of triangle do A(−2, 0), B(−4, 3) and C(−1, 6) make?
- A. Neither right-angled nor isosceles
- B. Right-angled, not isosceles
- C. Isosceles, no right angle
- D. Right-angled and isosceles
Choose the right answer
Answer:

13.A plaza has corners at A(0, 2), B(2, −2), C(−6, 5) and D(−8, 9). AC's mid-point = (, ) and BD's = (, ); the diagonals share their mid-point and bisect each other, so ABCD is a parallelogram.
Fill in each blank
Working:
Answer:
14.A triangular flower bed has corners at A(1, 0), B(4, 3) and C(10, −3). Prove ∠ABC = 90° from its sides' lengths: put the steps in order.
- A. = AC²
- B. ∠ABC = 90° (converse of Pythagoras' theorem)
- C. = 90
- D. AB² + BC² = 18 + 72
Write the letters in the right order
Working:
Answer:
15.A field has corners at A(−3, −3), B(−4, −4) and C(−6, −2). To prove ∠ABC is a right angle: AB's slope = and BC's = ; they multiply to −1, so AB ⊥ BC and ∠ABC = 90°.
Fill in each blank
Working:
Answer:
16.A triangular roof frame has its top at A(1, −4) and its foot corners at B(2, −2) and C(3, −3). AB² = and AC² = ; so AB = AC and △ABC is isosceles.
Fill in each blank
Working:
Answer:
17.A triangular flower bed has corners at A(−1, 2), B(2, 3) and C(3, 0). Prove ∠ABC = 90° from its sides' lengths: put the steps in order.
- A. AB² + BC² = 10 + 10
- B. ∠ABC = 90° (converse of Pythagoras' theorem)
- C. = 20
- D. = AC²
Write the letters in the right order
Working:
Answer:
18.A plaza has corners at A(6, −1), B(−4, 2), C(5, 5) and D(15, 2). AC's mid-point = (, ) and BD's = (, ); the diagonals share their mid-point and bisect each other, so ABCD is a parallelogram.
Fill in each blank
Working:
Answer:

Show the answers and working

Proofs with coordinates worksheet: Answers and working

1. A → B → D → C
Add the two shorter sides' squares to get 10, which is AC²; the converse then gives the right angle.
2. 5, 5, 10
Squared distances: AB² = 5, BC² = 5, AC² = 10, and 5 + 5 = 10. By the converse of Pythagoras' theorem, ∠ABC = 90°.
3. D → A → B → C
Add the two shorter sides' squares to get 65, which is AC²; the converse then gives the right angle.
4. 10, 40, 50
Squared distances: AB² = 10, BC² = 40, AC² = 50, and 10 + 40 = 50. By the converse of Pythagoras' theorem, ∠ABC = 90°.
5. 14, −4
(−2 + x2, 3 + y2) = (6 + (6)2, −3 + (2)2): x = 6 + (6) − (−2) = 14, y = −4. Or: walk from C as B walks to A, and you reach D.

6. A. AB's slope = CD's slope
Parallel lines have equal slopes: work out AB's and CD's, and equal slopes prove AB ∥ CD.
7. 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.
8. 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.
9. C. AB's slope = CD's slope
Parallel lines have equal slopes: work out AB's and CD's, and equal slopes prove AB ∥ CD.
10. True
See whether AB and DC, and AD and BC, have equal slopes; or whether AC and BD share their mid-point.
11. False
See whether AB and DC, and AD and BC, have equal slopes; or whether AC and BD share their mid-point.
12. A. Neither right-angled nor isosceles
AB² = 13, BC² = 18, AC² = 37: two equal make it isosceles; the two smaller adding to the largest make a right angle.

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

13. −3, 72, −3, 72
AC: (0 + (−6)2, 2 + (5)2) = (−31, 72); BD: (2 + (−8)2, −2 + (9)2), the same.
14. D → C → A → B
AB² = 18 and BC² = 72 add to 90, which is AC²; the sides fit Pythagoras, so by the converse ∠ABC = 90°.
15. 1, −1
AB: −3 − (−4)−3 − (−4) = 11; BC: −2 − (−4)−6 − (−4) = −11; 11 × (−11) = −1.
16. 5, 5
AB² = (2 − (1))² + (−2 − (−4))² = 5; AC² = (3 − (1))² + (−3 − (−4))² = 5.
17. A → C → D → B
AB² = 10 and BC² = 10 add to 20, which is AC²; the sides fit Pythagoras, so by the converse ∠ABC = 90°.
18. 112, 2, 112, 2
AC: (6 + (5)2, −1 + (5)2) = (112, 21); BD: (−4 + (15)2, 2 + (2)2), the same.

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