Coordinates are a brilliant system for pinpointing any location using just two numbers. You already know the first quadrant from earlier years — in Grade 6, we extend to ALL FOUR quadrants, using negative numbers. You'll also find midpoints, describe translations using coordinates, and use coordinates to describe shapes.
📍 Reading Coordinates
(x, y) — along the corridor, then up/down the stairs! x is horizontal, y is vertical
📐 Four Quadrants
Q1: (+,+) · Q2: (−,+) · Q3: (−,−) · Q4: (+,−)
📏 Midpoint Formula
Midpoint of (2,4) and (8,2) = ((2+8)÷2, (4+2)÷2) = (5, 3)
The Four Quadrants
The x-axis and y-axis divide the grid into four quadrants. Here is how they are arranged:
y-axis ↑
Q2 (−, +)
Q1 (+, +)
Q3 (−, −)
Q4 (+, −)
← x-axis →
⭐ Golden Rules
Always read x FIRST (horizontal), then y (vertical). Remember: "along the corridor, then up the stairs."
The x-axis is the horizontal line. The y-axis is the vertical line. They cross at the ORIGIN (0, 0).
Negative x: go LEFT of the origin. Negative y: go DOWN from the origin.
Four quadrants: Q1 (+,+), Q2 (−,+), Q3 (−,−), Q4 (+,−). Go anti-clockwise!
Midpoint formula: midpoint = ((x₁+x₂)/2, (y₁+y₂)/2). Average the x-values and the y-values separately.
To describe a translation using coordinates: if a point moves from (2,3) to (5,7), the translation is (+3, +4) — add 3 to x, add 4 to y.
Coordinates of shape vertices: when given vertices, you can check if it's a square/rectangle etc. by measuring distances between points.
📖 Worked Examples
Example 1: Reading and Plotting in All 4 Quadrants
Identify which quadrant each point belongs to:
📍 (3, 5) → x is positive, y is positive → Q1 (+, +)
📍 (−2, 4) → x is negative, y is positive → Q2 (−, +)
📍 (−3, −1) → x is negative, y is negative → Q3 (−, −)
📍 (4, −2) → x is positive, y is negative → Q4 (+, −)
Tip: Check the sign of x first, then the sign of y. That tells you the quadrant straight away!
Example 2: Finding the Midpoint
Find the midpoint of A(−4, 2) and B(6, −8).
Step 1: Average the x-values → (−4 + 6) ÷ 2 = 2 ÷ 2 = 1
Step 2: Average the y-values → (2 + (−8)) ÷ 2 = −6 ÷ 2 = −3
✅ Midpoint = (1, −3)
Remember: The midpoint is exactly halfway between the two points. Add the coordinates and divide by 2 — separately for x and y.
Example 3: Coordinates of a Shape
A rectangle has vertices at A(−3, 2), B(4, 2), C(4, −1), D(−3, −1).
Step 1: Find the length AB (horizontal side, same y-value) → 4 − (−3) = 7 units
Step 2: Find the width BC (vertical side, same x-value) → 2 − (−1) = 3 units
✅ Area = 7 × 3 = 21 square units
Tip: For horizontal distances, subtract the x-values. For vertical distances, subtract the y-values. Always subtract the smaller from the larger (or use absolute value).
Example 4: Translation using Coordinates
Point P(2, −3) is translated by the vector (+4, +5). Find the new position.
Step 1: New x = 2 + 4 = 6
Step 2: New y = −3 + 5 = 2
✅ New position: P'(6, 2)
Remember: A translation adds the vector to each coordinate. A positive number moves right/up; a negative number moves left/down.
🔬 Coordinate Plotter
Enter coordinates to plot points on the grid. Plot two points to see the midpoint automatically!
How to use this plotter:
Type any x and y value between −8 and 8 (decimals allowed!)
Click Plot Point to add it to the grid
After plotting 2 or more points, the midpoint of the last two is calculated automatically
A dashed line connects the last two points, and the midpoint is shown as a pink dot
Click Clear to start again
🎯 Drag Exercise 1: Identify the Quadrant
In which quadrant is the point (−3, 5)?
Drag the correct answer into the drop zone!
Q1
Q2
Q3
Q4
(−3, 5) belongs in quadrant →
?
💡 Hint: x = −3 is negative → LEFT of the origin. y = 5 is positive → ABOVE the origin. Which quadrant is top-left?
🎯 Drag Exercise 2: Reading Coordinates
A point is 4 units left of the origin and 2 units below. What are its coordinates?
Drag the correct answer into the drop zone!
(4,-2)
(-4,2)
(-4,-2)
(4,2)
Coordinates →
?
💡 Hint: Left of origin = negative x. Below the origin = negative y.
🎯 Drag Exercise 3: Midpoint
Find the midpoint of (2, 6) and (8, 2).
Drag the correct answer into the drop zone!
(4,3)
(5,4)
(5,3)
(6,4)
Midpoint →
?
💡 Hint: Average the x-values: (2 + 8) ÷ 2 = ? Average the y-values: (6 + 2) ÷ 2 = ?
🎯 Drag Exercise 4: Translation
Point A is at (−2, 3). It is translated by (+5, −4). Where is the new point A'?
Drag the correct answer into the drop zone!
(3,-1)
(3,7)
(-7,-1)
(7,-1)
New point A' →
?
💡 Hint: Add 5 to x: −2 + 5 = ? Add −4 to y: 3 + (−4) = ?
🎯 Drag Exercise 5: Distance on a Grid
Point A is at (−3, 2) and B is at (4, 2). What is the length of AB?
Drag the correct answer into the drop zone!
5
6
7
8
Length of AB →
?
💡 Hint: Both points have the same y-value (y = 2), so this is a horizontal line. Count along the x-axis: 4 − (−3) = ?
🎯 Drag Exercise 6: Missing Vertex
Three vertices of a rectangle are A(1, 4), B(5, 4), C(5, −2). What are the coordinates of the 4th vertex D?
Drag the correct answer into the drop zone!
(1,-2)
(-1,2)
(1,2)
(-1,-2)
4th vertex D →
?
💡 Hint: D must be directly below A and directly to the left of C. A has x = 1, and C has y = −2. So D has x = 1 and y = −2.
✏️ Practice Questions
Answer all 20 questions, then reveal the answers to check your work!
Write the coordinates of a point 3 right and 4 up from the origin.
Write the coordinates of a point 2 left and 5 down.
In which quadrant is (−4, −7)?
In which quadrant is (6, −3)?
Find the midpoint of (0, 0) and (8, 4).
Find the midpoint of (−6, 4) and (2, −2).
A point at (3, 5) is translated by (−4, +2). Find the new coordinates.
A point at (−2, −3) is translated by (+5, +6). Find the new coordinates.
What is the y-coordinate of any point on the x-axis?
What is the x-coordinate of any point on the y-axis?
Find the length of a horizontal line segment from (−3, 2) to (5, 2).
Find the length of a vertical line segment from (4, −3) to (4, 6).
A square has two vertices at (0, 0) and (4, 0). What could the other two vertices be?
Find the midpoint of (−3, −5) and (3, 5).
Translate the point (2, −4) by (−3, −1).
A point is in Q3. Are both its coordinates negative or positive?
What is the distance from (−5, 0) to (3, 0)?
Find the coordinates of the midpoint of a line from (7, −3) to (1, 9).
A triangle has vertices A(0, 0), B(6, 0), C(3, 4). Find the midpoint of AC.
A point moves from (2, 3) to (−1, 7). Describe the translation.
✅ Answers
(3, 4)
(−2, −5)
Q3 (both coordinates are negative)
Q4 (x is positive, y is negative)
(4, 2) — midpoint x = (0+8)÷2 = 4; midpoint y = (0+4)÷2 = 2
(−2, 1) — midpoint x = (−6+2)÷2 = −2; midpoint y = (4+(−2))÷2 = 1
(−1, 7) — new x = 3+(−4) = −1; new y = 5+2 = 7
(3, 3) — new x = −2+5 = 3; new y = −3+6 = 3
0 — every point on the x-axis has y = 0
0 — every point on the y-axis has x = 0
8 units — 5 − (−3) = 8
9 units — 6 − (−3) = 9
(4, 4) and (0, 4) (or (4, −4) and (0, −4) for a square below the x-axis)
(0, 0) — the origin! midpoint x = (−3+3)÷2 = 0; midpoint y = (−5+5)÷2 = 0
(−1, −5) — new x = 2+(−3) = −1; new y = −4+(−1) = −5
Both negative — Q3 is the region where x < 0 and y < 0
8 units — both points are on the x-axis: 3 − (−5) = 8
(4, 3) — midpoint x = (7+1)÷2 = 4; midpoint y = (−3+9)÷2 = 3
(1.5, 2) — midpoint x = (0+3)÷2 = 1.5; midpoint y = (0+4)÷2 = 2
(−3, +4) — change in x = −1−2 = −3; change in y = 7−3 = +4
🏆 Challenge Problems
These are harder problems — take your time and show your working!
The midpoint of AB is (3, −1). A is at (7, 3). Find the coordinates of B.
Three vertices of a parallelogram are A(0, 0), B(4, 0), C(6, 3). Find the coordinates of D so that ABCD is a parallelogram.
A square has vertices at (1, 1), (4, 1), (4, 4) and (1, 4). Find the midpoint of each diagonal. Are they the same? What does this tell you?
A line passes through (0, 3) and (6, 0). Where does it cross each axis? Find the midpoint of the segment between the two axis-crossing points.
Point P has coordinates (x, 4). P is equidistant from A(1, 0) and B(7, 8). Find the value of x.
A rectangle ABCD has A(−3, 4), B(5, 4), C(5, −2). Find D and calculate the area of the rectangle.
Starting at (0, 0), a robot moves: right 3, up 4, left 5, down 2. What are the final coordinates?
Four points form a shape: A(0, 4), B(3, 0), C(0, −4), D(−3, 0). Find the length of each side. What type of shape is it?
The coordinates of a shape's vertices are A(1, 5), B(4, 9), C(7, 5), D(4, 1). Find the midpoints of both diagonals. What type of shape is it?
A ship starts at (−4, −3). It travels to (2, 5), then to (8, 1). What is the total distance travelled? Give your answer to 1 decimal place.
✅ Challenge Answers
B = (−1, −5)
If midpoint = (3, −1) and A = (7, 3), then:
Bx = 2 × 3 − 7 = 6 − 7 = −1
By = 2 × (−1) − 3 = −2 − 3 = −5
D = (2, 3)
In parallelogram ABCD, D = A + C − B = (0 + 6 − 4, 0 + 3 − 0) = (2, 3) Check: AB is parallel and equal to DC — both go (+4, 0).
Both midpoints = (2.5, 2.5)
Diagonal 1: midpoint of (1,1) and (4,4) = ((1+4)÷2, (1+4)÷2) = (2.5, 2.5)
Diagonal 2: midpoint of (4,1) and (1,4) = ((4+1)÷2, (1+4)÷2) = (2.5, 2.5) They are the same — the diagonals of a square bisect each other.
Crosses y-axis at (0, 3); crosses x-axis at (6, 0). Midpoint = (3, 1.5)
Midpoint x = (0+6)÷2 = 3; Midpoint y = (3+0)÷2 = 1.5
x = 4, so P = (4, 4)
PA² = (x−1)² + (4−0)² = (x−1)² + 16
PB² = (x−7)² + (4−8)² = (x−7)² + 16
Setting equal: (x−1)² = (x−7)²
x² − 2x + 1 = x² − 14x + 49
12x = 48, so x = 4
D = (−3, −2); Area = 48 square units
D has the same x as A (x = −3) and same y as C (y = −2), so D = (−3, −2)
Width = 5 − (−3) = 8; Height = 4 − (−2) = 6; Area = 8 × 6 = 48
(−2, 2)
Right 3: x = 3 | Up 4: y = 4 | Left 5: x = 3−5 = −2 | Down 2: y = 4−2 = 2
Final position: (−2, 2)
All sides = 5 units; the shape is a rhombus
AB = √((3−0)² + (0−4)²) = √(9+16) = √25 = 5
BC = √((0−3)² + (−4−0)²) = √(9+16) = 5
CD = √((−3−0)² + (0−(−4))²) = 5 | DA = 5
All sides equal. Diagonals: AC = 8, BD = 6 (not equal) → rhombus
Both diagonal midpoints = (4, 5); the shape is a rhombus
Diagonal AC: midpoint of (1,5) and (7,5) = (4, 5)
Diagonal BD: midpoint of (4,9) and (4,1) = (4, 5) ✓
All sides: AB = √(9+16) = 5; BC = √(9+16) = 5; CD = 5; DA = 5 (all equal)
Diagonals: AC = 6, BD = 8 (not equal) → rhombus (not a square)
Total ≈ 17.2 units
Leg 1: (−4,−3) to (2,5) → √((2−(−4))²+(5−(−3))²) = √(36+64) = √100 = 10
Leg 2: (2,5) to (8,1) → √((8−2)²+(1−5)²) = √(36+16) = √52 ≈ 7.2
Total = 10 + 7.2 = 17.2 units