Coordinate Geometry

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टीचर्स के लिए: Coordinate Geometry (Maths [CIE], Extended) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके स्टूडेंट्स लाइव गेम की तरह खेलें।

लेसन नोट्स

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

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 48 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

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

और सवाल

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

रिवीज़न नोट्स

## 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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