A(6, 7), B(5, 5), C(1, 7). AB^2 = □, BC^2 = □, AC^2 = □; since AB^2 + BC^2 = AC^2, ∠ABC = 90°. Fill in each blank

5, 20, 25. Squared distances: AB^2 = 5, BC^2 = 20, AC^2 = 25, and 5 + 20 = 25. By the converse of Pythagoras' theorem, ∠ABC = 90°.

A(5, −5), B(3, 3), C(2, 2), D(4, −6). ABCD is a parallelogram. True or false?

True. See whether AB and DC, and AD and BC, have equal slopes; or whether AC and BD share their mid-point.

ABCD is a parallelogram with A(−4, 0), B(4, −6), C(0, 0). Its diagonals bisect each other, so AC's mid-point is BD's, and D = (□, □). Fill in each blank

−8, 6. ((4 + x)/2, (−6 + y)/2) = ((−4 + (0))/2, (0 + (0))/2): x = −4 + (0) − (4) = −8, y = 6. Or: walk from C as B walks to A, and you reach D.

Coordinate geometry · Ages 14–15

Proofs with coordinates

Right angle: product −1 or the converse; parallel: equal slopes.

How to do it

Method

Right angle: product −1 or the converse; parallel: equal slopes

12 × (−2) = −1, so AB ⊥ BC

Another way

By the lengths

AB² + BC² = 20 + 20 = 40 = AC²

In full

To prove a figure's property with coordinates, decide what to work out, then write the working and the conclusion. ① A right angle: find two sides' slopes; if they multiply to −1, the sides are perpendicular. Or by lengths: find the three sides, and if AB² + BC² = AC², by the converse of Pythagoras' theorem ∠ABC = 90°. Both ways are right. ② Parallel: the two segments' slopes are equal. ③ A parallelogram: both pairs of opposite sides have equal slopes (both pairs parallel); or the two diagonals share their mid-point (the diagonals bisect each other). ④ An isosceles triangle: by the distance formula, two sides are equally long. A(0, 0), B(4, 2), C(2, 6): AB's slope 24 = 12, BC's 4−2 = −2, 12 × (−2) = −1, so AB ⊥ BC and ∠ABC = 90°; or AB² = 20, BC² = 20, AC² = 40, and 20 + 20 = 40. The last line states the conclusion and its reason.

Worked examples

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

Example 1
A(6, 7), B(5, 5), C(1, 7). AB² = , BC² = , AC² = ; since AB² + BC² = AC², ∠ABC = 90°.
Fill in each blank
Show the answer and working
Answer: 5, 20, 25
Squared distances: AB² = 5, BC² = 20, AC² = 25, and 5 + 20 = 25. By the converse of Pythagoras' theorem, ∠ABC = 90°.
Example 2
A(5, −5), B(3, 3), C(2, 2), D(4, −6). ABCD is a parallelogram.
True or false?
Show the answer and working
Answer: True
See whether AB and DC, and AD and BC, have equal slopes; or whether AC and BD share their mid-point.
Example 3
ABCD is a parallelogram with A(−4, 0), B(4, −6), C(0, 0). Its diagonals bisect each other, so AC's mid-point is BD's, and D = (, ).
Fill in each blank
Show the answer and working
Answer: −8, 6
(4 + x2, −6 + y2) = (−4 + (0)2, 0 + (0)2): x = −4 + (0) − (4) = −8, y = 6. Or: walk from C as B walks to A, and you reach D.

Practice questions to try

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

1.A(4, −6), B(−4, −1), C(0, −5), D(−2, 6). 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
- C. AB = CD
- D. AB's slope × CD's slope = −1
Choose the right answer
Answer:
2.A(−5, −6), B(−3, 5), C(6, 4), D(4, −7). ABCD is a parallelogram.
True or false?
Answer:
3.A(5, 8), B(4, 5), C(−2, 7). Prove ∠ABC = 90° by the lengths: put the steps in order.
- A. ∠ABC = 90° (converse of Pythagoras' theorem)
- B. = AC²
- C. = 50
- D. AB² + BC² = 10 + 40
Write the letters in the right order
Answer:
4.ABCD is a parallelogram with A(5, −2), B(5, −5), C(−2, −4). Its diagonals bisect each other, so AC's mid-point is BD's, and D = (, ).
Fill in each blank
Answer:
5.What kind of triangle do A(4, 0), B(1, 0) and C(1, 3) make?
- A. Neither right-angled nor isosceles
- B. Isosceles, no right angle
- C. Right-angled and isosceles
- D. Right-angled, not isosceles
Choose the right answer
Answer:

All the answers

1. B. AB's slope = CD's slope
2. True
3. D → C → B → A
4. −2, −1
5. C. Right-angled and isosceles

Free worksheet: Proofs with coordinatesWorking 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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