Coordinates
Plotting points and finding areas on the coordinate plane, transformations of a point, and the distance, mid-point and slope formulas, including parallel and perpendicular lines. They are used again in straight lines, loci and circles.
What you will learn
- 26.1–26.3 Plot points; distances along horizontal and vertical lines; areas of polygons on the plane.
- 26.4 Transformations of a point:
- 26.5 Distance formula.
- 26.6 Mid-point formula; NF internal point of division.
- 26.7 Slope formula; intercepts; find a missing coordinate from a slope.
- 26.8 Parallel lines (equal slopes); perpendicular lines (slopes multiply to −1).
- 26.9 Simple proofs with coordinates (for example, show a triangle is right-angled).
You need first: Geometry and data basics, Surds
Key terms
| English | 中文 |
|---|---|
| rectangular coordinate plane | 直角坐標平面 |
| x-coordinate, y-coordinate | x 坐標，y 坐標 |
| origin | 原點 |
| distance formula | 距離公式 |
| mid-point formula | 中點公式 |
| point of division | 內分點 |
| slope | 斜率 |
| x-intercept, y-intercept | x 截距，y 截距 |
| translation, reflection, rotation | 平移，反射，旋轉 |
| clockwise / anticlockwise | 順時針／逆時針 |
| collinear | 共線 |
Standard steps
The formulas, for A(x₁, y₁) and B(x₂, y₂)
| Find | Formula | Why |
|---|---|---|
| Distance AB | √[(x₂ − x₁)² + (y₂ − y₁)²] | Pythagoras on the horizontal and vertical gaps |
| Mid-point | ((x₁ + x₂)/2, (y₁ + y₂)/2) | the average of the ends |
| Slope | (y₂ − y₁)/(x₂ − x₁) | rise ÷ run |
| Point dividing AB in r : s (NF) | ((s·x₁ + r·x₂)/(r + s), (s·y₁ + r·y₂)/(r + s)) | weighted average, closer to A when r is small |
- Label the points A(x₁, y₁) and B(x₂, y₂).
- Write the formula, then substitute. Put negative coordinates in brackets: 4 − (−2).
- Simplify. Leave a distance as an exact surd unless the question asks for a decimal.
- For a missing coordinate, substitute the known slope or distance and solve the equation.
Rotation about the origin
| Rotation | (x, y) becomes |
|---|---|
| 90° anticlockwise | (−y, x) |
| 180° | (−x, −y) |
| 90° clockwise | (y, −x) |
- Sketch the point and decide which quadrant it moves to.
- Apply the rule from the table.
- Check that the answer lies in the quadrant from step 1.
Area of a triangle from coordinates
- Draw the smallest rectangle around the triangle, with sides parallel to the axes.
- Find the area of each right-angled triangle in the corners: ½ × run × rise, read from the coordinates.
- Area of the triangle = rectangle − the corner triangles.
Show three points are collinear
- Find the slope of AB.
- Find the slope of BC.
- If the slopes are equal, AB and BC are parallel and share the point B, so they are one line: A, B and C are collinear.
Worked examples
Example 1: Use the formulas for A(−2, 1) and B(4, 9): distance, mid-point and slope.
AB = √[(4 − (−2))² + (9 − 1)²] = √(36 + 64) = 10
mid-pt = ((−2 + 4)/2, (1 + 9)/2) = (1, 5)
slope = (9 − 1)/(4 − (−2)) = 8/6 = 4/3
a line ⊥ AB has slope −3/4 (4/3 × −3/4 = −1)Example 2: NF: P divides AB in the ratio AP : PB = 1 : 3.
P = ((3(−2) + 1(4))/4, (3(1) + 1(9))/4) = (−1/2, 3)Example 3: Rotation about the origin: rotate (3, −1) through 90° anticlockwise about O.
(3, −1) is in quadrant IV; a quarter-turn anticlockwise takes it to quadrant I
(x, y) → (−y, x)
(3, −1) → (1, 3) quadrant I, as expectedExample 4: Area of a triangle from coordinates: vertices (1, 1), (7, 3), (3, 6).
rectangle 6 × 5 = 30
corner triangles: ½·6·2 + ½·4·3 + ½·2·5 = 6 + 6 + 5 = 17
area = 30 − 17 = 13Example 5: The line through A(1, k) and B(5, 7) has slope 1/2. Find k, then the slope of a line perpendicular to AB.
slope AB = (7 − k)/(5 − 1) = 1/2
7 − k = 2
k = 5
perpendicular slope = −1 ÷ (1/2) = −2 (1/2 × −2 = −1)Example 6: Show that the three points P(−1, −1), Q(1, 3) and R(3, 7) are collinear.
slope PQ = (3 − (−1))/(1 − (−1)) = 4/2 = 2
slope QR = (7 − 3)/(3 − 1) = 4/2 = 2
slope PQ = slope QR and Q is a common point
∴ P, Q and R are collinearCommon mistakes
- Subtracting the coordinates in different orders on the top and bottom of the slope formula.
- Sign errors with negative coordinates: 4 − (−2) = 6, not 2.
- In the point-of-division formula, pairing r with x₁ instead of x₂.
- Rotating clockwise when the question says anticlockwise.
- Perpendicular slope written as 3/4 (sign forgotten) or −4/3 (not flipped).
How the DSE tests it
- A(1): transform a point (rotate, reflect, translate), then test collinearity or perpendicularity by slopes.
- A(2) and Section B: these formulas inside loci and circle questions, often with an exact surd distance.
- Paper 2: mid-point, distance and slope items, and the point of division (NF).
Questions you might not ask in class
Does it matter which point is (x₁, y₁)?
No, as long as you keep the same order on the top and the bottom.
Why do perpendicular slopes multiply to −1?
Rotating a line 90° turns "rise 4, run 3" into "rise −3, run 4". The new slope is −3/4, and 4/3 × −3/4 = −1.
What does a slope of 0 or 'undefined' mean?
0 is a horizontal line. Undefined (run = 0) is a vertical line.