Coordinate Geometry

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Coordinates

The Cartesian plane has a horizontal x-axis and vertical y-axis meeting at the origin (0,0).
Coordinates are written as (x, y) – x is horizontal, y is vertical.
Positive x: right of origin; negative x: left. Positive y: above; negative y: below.
Mnemonic: "Along the corridor, up the stairs" (x then y).
Check the scale on axes – 1 square may not equal 1 unit.

Points on the Cartesian plane

Midpoint of a Line

The midpoint is the average (mean) of the endpoints' coordinates.
Formula: midpoint = ( (x₁+x₂)/2 , (y₁+y₂)/2 ).
Example: A(-4,3) and B(8,-12) → midpoint = (2, -4.5).

Midpoint of AB

Gradient of a Line

Gradient measures steepness: rise/run (change in y over change in x).
Positive gradient: line goes uphill (bottom-left to top-right).
Negative gradient: line goes downhill (top-left to bottom-right).
Formula: gradient m = (y₂ - y₁)/(x₂ - x₁).
To draw a gradient, e.g. 2/3: move 3 right, 2 up. For -5: move 1 right, 5 down.
A gradient of 3 is steeper than 2; -5 is steeper than -4.

Gradient as rise over run

Length of a Line

Distance between two points uses Pythagoras' theorem.
Formula: d = √[(x₁ - x₂)² + (y₁ - y₂)²].
Be careful with negative coordinates – use brackets to avoid errors.
Example: A(3,-4) and B(-5,2) → d = √(8² + (-6)²) = √100 = 10 units.

Length of AB

Slides

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Practice questions

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1.What is the midpoint of the line segment joining A(1, 3) and B(5, 8)?
Easy
- A(3, 5.5)
- B(2, 2.5)
- C(6, 11)
- D(4, 5)
2.The point A has coordinates (5, −4). The point B has coordinates (13, 1). Work out the coordinates of the midpoint of AB.
Easy
- A(9, -1.5)
- B(8, -3)
- C(18, -3)
- D(9, 2.5)
3.Find the gradient of the straight line with equation 5x + 2y = 7.
Medium
- A-5/2
- B5/2
- C-2/5
- D2/5
4.Line L has equation y = 2 − 3x. Write down the gradient of line L.
Easy
- A-3
- B3
- C2
- D-2
5.Find the mid-point of AB where A = (w, r) and B = (3w, t). Give your answer in its simplest form in terms of w, r and t.
Medium
- A(2w, (r+t)/2)
- B(4w, r+t)
- C(2w, r+t)
- D(w, (r+t)/2)
6.A is the point (2, 3) and B is the point (7, −5). Find the coordinates of the midpoint of AB.
Easy
- A(4.5, -1)
- B(9, -2)
- C(4.5, 1)
- D(5, -2)
7.Point A has coordinates (5, 8) and point B has coordinates (9, −4). Work out the gradient of AB.
Easy
- A-3
- B3
- C-1/3
- D1/3
8.A straight line joins the points (3k, 6) and (k, −5). The line has a gradient of 2. Find the value of k.
Medium
- A11/2
- B11/4
- C11
- D-11/2

More questions

9.P is the point (16, 9) and Q is the point (22, 24). N is the point on PQ such that PN = 2NQ. Find the coordinates of N.
Hard
- A(20, 19)
- B(18, 14)
- C(19, 17)
- D(20, 14)
10.A and B are two points. Point A has coordinates (−2, 4). Point B has coordinates (8, 9). C is the midpoint of the line segment AB. Find the coordinates of C.
Easy
- A(3, 6.5)
- B(6, 13)
- C(3, 6)
- D(5, 5)
11.D is the point with coordinates (100, 56). Does point D lie on the straight line that passes through A(−2, 4) and B(8, 9)? You must show how you work out your answer.
Hard
- AYes, because gradient AB = 0.5 and line equation gives y=56 when x=100
- BNo, because gradient AB = 0.5 and line equation gives y=55 when x=100
- CYes, because D is collinear with A and B
- DNo, because the line does not pass through (100,56)
12.The points A, B and C lie in order on a straight line. The coordinates of A are (2, 5). The coordinates of B are (4, p). The coordinates of C are (q, 17). Given that AC = 4AB, find the values of p and q.
Hard
- Ap = 8, q = 10
- Bp = 6, q = 12
- Cp = 7, q = 11
- Dp = 9, q = 9
13.A and B are straight lines. Line A has equation 2y = 3x + 8. Line B goes through the points (−1, 2) and (2, 8). Do lines A and B intersect? You must show all your working.
Hard
- AYes, at (0,4)
- BNo, they are parallel
- CYes, at (2,8)
- DNo, they are the same line
14.Point A has coordinates (2, 3). Point B has coordinates (6, 8). M is the midpoint of the line AB. Find the coordinates of M.
Easy
- A(4, 5.5)
- B(8, 11)
- C(4, 5)
- D(3, 4)
15.The straight line L has equation 2y + 7x = 10. Find the gradient of L.
Medium
- A-7/2
- B7/2
- C-2/7
- D2/7
16.A point has coordinates (−2, 5). What are the coordinates of the point that is 3 units to the right and 4 units down?
Easy
- A(1, 1)
- B(-5, 9)
- C(1, 9)
- D(-5, 1)
17.What is the midpoint of the line segment joining (0, 0) and (8, −6)?
Easy
- A(4, -3)
- B(-4, 3)
- C(4, 3)
- D(8, -6)
18.Find the gradient of the line through points (1, 2) and (4, 8).
Easy
- A2
- B1/2
- C6
- D3
19.A line has equation y = 4x − 3. What is its gradient?
Easy
- A4
- B-3
- C3
- D-4
20.Find the length of the line segment joining (0, 0) and (3, 4).
Easy
- A5
- B7
- C1
- D25
21.The midpoint of AB is (4, 7). A is (2, 3). What are the coordinates of B?
Medium
- A(6, 11)
- B(3, 5)
- C(8, 14)
- D(0, -1)
22.Find the gradient of the line 3x + 4y = 12.
Medium
- A-3/4
- B3/4
- C-4/3
- D4/3
23.Calculate the length of AB given A(1, 2) and B(4, 6).
Easy
- A5
- B√13
- C√45
- D3
24.A line passes through (−1, 5) and (3, 1). What is its gradient?
Easy
- A-1
- B1
- C-2
- D2
25.Point P is the midpoint of A(2, 8) and B(6, 2). What are the coordinates of P?
Easy
- A(4, 5)
- B(8, 10)
- C(3, 4)
- D(4, 3)
26.Points A(1, 2), B(4, 6), and C(7, 10) lie on a straight line. What is the gradient of the line?
Easy
- A4/3
- B3/4
- C2
- D1
27.A line has gradient −2 and passes through (3, 4). What is its equation in the form y = mx + c?
Medium
- Ay = -2x + 10
- By = -2x - 2
- Cy = 2x - 2
- Dy = -2x + 4
28.The length of the line segment joining (a, 0) and (0, a) is 5√2. Find a positive value of a.
Medium
- A5
- B10
- C√5
- D2.5
29.Which of the following points lies on the line y = 3x − 2?
Easy
- A(2, 4)
- B(2, 3)
- C(1, 4)
- D(0, 3)
30.Find the midpoint of the line segment joining (−3, 7) and (5, −1).
Easy
- A(1, 3)
- B(-1, 4)
- C(2, 6)
- D(4, -4)
31.What is the x-coordinate of the point that is 4 units to the left of the origin?
Easy
- A-4
- B4
- C0
- D1
32.The point (3, -2) is located 3 units to the right of the origin and 2 units down.
Easy
True or false?
33.What is the midpoint of the line segment joining (0, 0) and (6, 8)?
Easy
- A(3, 4)
- B(2, 4)
- C(3, 8)
- D(6, 8)
34.A line has gradient -3. Which of the following could be the coordinates of two points on the line?
Medium
- A(0,0) and (1,-3)
- B(0,0) and (1,3)
- C(0,0) and (2,3)
- D(0,0) and (3,1)
35.The midpoint of AB is (3, 5). A is (1, 2). What are the coordinates of B?
Medium
- A(5, 8)
- B(2, 3)
- C(4, 7)
- D(5, 3)
36.The line with equation y = 2x + 1 has a gradient of 2.
Easy
True or false?
37.The gradient of the line joining (2k, 5) and (k, 1) is 2. Find k.
Medium
- A2
- B-2
- C4
- D-4
38.Match each gradient description with its value.
Medium
- Uphill, 2 units up for 1 right
- Downhill, 3 down for 1 right
- Horizontal line
- Vertical line
- 2
- -3
- 0
- undefined
39.Arrange the steps to find the midpoint of A(2, 5) and B(6, 9) in correct order.
Easy
- Add the x-coordinates: 2 + 6 = 8
- Add the y-coordinates: 5 + 9 = 14
- Divide each sum by 2: (8/2, 14/2)
- Write the midpoint: (4, 7)
40.Which of the following points lie on the line that passes through (1, 2) and (3, 6)? Select all that apply.
Medium
- A(0, 0)
- B(2, 4)
- C(4, 8)
- D(5, 9)
- E( -1, -2)
41.Complete the sentence.
Easy
The coordinates of the origin are (____, ____).
42.Find the gradient of the line through points (1, 2) and (3, 6).
Easy
43.Calculate the length of the line segment joining A(1, 2) and B(4, 6).
Medium
44.Point A has coordinates (2, 3) and point B has coordinates (5, 7). Find the length of AB.
Medium
45.Complete the sentence about gradients.
Easy
A line with gradient ____ is steeper than a line with gradient 2.
46.Find the gradient of the line through points (1, 2) and (3, 6).
Easy
- A1
- B2
- C4
- D3
47.Calculate the length of the line segment joining A(1, 2) and B(4, 6).
Medium
- A4 units
- B7 units
- C6 units
- D5 units
48.Point A has coordinates (2, 3) and point B has coordinates (5, 7). Find the length of AB.
Medium
- A5
- B4
- C7
- D6

Revision notes

## Coordinate Geometry - Quick Revision ### Coordinates - Cartesian plane: x-axis (horizontal), y-axis (vertical), origin (0,0). - Coordinates are (x, y): "along the corridor, up the stairs". ### Midpoint of a Line - Midpoint = average of coordinates: $$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$ ### Gradient of a Line - Gradient = steepness: $\frac{\text{change in }y}{\text{change in }x} = \frac{\text{rise}}{\text{run}}$. - Between two points $(x_1,y_1)$ and $(x_2,y_2)$: $$\frac{y_2-y_1}{x_2-x_1}$$ - Positive gradient: uphill (left to right). Negative: downhill. - To draw gradient $\frac{a}{b}$: move $b$ right, $a$ up (if positive) or down (if negative). ### Length of a Line - Distance between $(x_1,y_1)$ and $(x_2,y_2)$: $$d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$$ - Uses Pythagoras' theorem. Be careful with negative coordinates. ### Key Tips - Check scale on axes. - For gradient, always check sign (uphill/downhill). - For length, square differences, add, then square root.

Coordinates: A pair of numbers (x, y) that describe the location of a point on the Cartesian plane, where x is horizontal distance from origin and y is vertical distance.
Midpoint: The point halfway between two points, found by averaging their x and y coordinates: ((x1+x2)/2, (y1+y2)/2).
Gradient: A measure of steepness of a line, calculated as change in y divided by change in x (rise/run). Positive for uphill, negative for downhill.
Length of a line: The distance between two points, calculated using Pythagoras: d = √((x1-x2)² + (y1-y2)²).

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