GCSE Maths: Coordinates in four quadrants (Foundation)
Part of the GCSE Maths straight-line graphs and coordinates chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Coordinates in four quadrants
A coordinate grid lets us give the exact position of any point using two numbers. It is made from two number lines that cross at right angles. The horizontal one is the x-axis and the vertical one is the y-axis. They cross at the origin, which is the point (0, 0).
A point is described by a pair of numbers in brackets, called its coordinates: (x, y). The first number says how far to go across from the origin, and the second says how far to go up or down. A good way to remember the order is “along the corridor, then up the stairs”: you cannot go up before you have gone along. In the alphabet x comes before y, which gives the same order.
Numbers on the axes carry on past zero into the negatives. Going right or up is positive. Going left or down is negative. So the point (−4, 3) is 4 squares to the left of the origin and 3 squares up.
The four regions are called quadrants. They are numbered anticlockwise, starting at the top right. The signs of the coordinates tell you which quadrant a point is in: in Quadrant 1 both coordinates are positive; in Quadrant 3 both are negative. A point on an axis has one coordinate equal to 0, so it is not inside any quadrant. Points on the y-axis have x = 0, for example (0, 4); points on the x-axis have y = 0, for example (−3, 0).
Coordinates and shapes
Corners of shapes are called vertices (one is a vertex). On a grid, points on a horizontal line all have the same y-coordinate, and points on a vertical line all have the same x-coordinate. This lets you find missing corners of rectangles, squares and parallelograms, and work out lengths by subtracting. The distance from −3 to 5 along an axis is 5 − (−3) = 8: always take the smaller number from the larger.
Key ideas
- The coordinates (x, y) are written across first, then up or down.
- The origin is (0, 0). Right and up are positive; left and down are negative.
- Quadrants are numbered 1, 2, 3, 4 anticlockwise from the top right: (+, +), (−, +), (−, −), (+, −).
- A point on an axis has one coordinate equal to 0.
- In a rectangle, opposite sides are equal and parallel; in a parallelogram too. Use this to find a missing vertex.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. For each point go across from the origin first, then up or down. Write the coordinates as (x, y).
- Point P: how far across? how far up or down? Across means along the x-axis. Right is positive.P is 3 right and 2 up: P = (3, 2)
- Point Q: left of the origin. Left is negative, so the x-coordinate is negative.Q is 4 left and 1 up: Q = (−4, 1)
- Point R: left and down. Both coordinates are negative.R is 2 left and 3 down: R = (−2, −3)
- Point S: it is on the y-axis. It is directly above the origin, so there is no movement across: x = 0.S is 0 across and 4 up: S = (0, 4)
Check. Start at (0, 0) and follow each pair of numbers. You arrive at the right point each time. ✓
Watch out. Across first, then up or down: (3, 2) is not the same as (2, 3).
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. In a rectangle, opposite sides are parallel and equal, so the move from B to A is the same as the move from C to D.
- Plot the three points A, B and C. A sketch helps you see where the fourth corner has to go. The corners are labelled in order round the shape.A = (−4, −2), B = (3, −2), C = (3, 1)
- Work out the move from B to A. In a rectangle, opposite sides are parallel and equal in length, and every angle is 90°. So side CD is the same move as side BA.across: −4 − 3 = −7 (7 left)
up/down: −2 − (−2) = 0 (0) - Start at C and make the same move to reach D. C is at (3, 1). Add the same across and up/down numbers.x: 3 + (−7) = −4
y: 1 + 0 = 1 - Write the coordinates and draw the shape. D = (−4, 1)
Check. The diagonals of a rectangle cross at their midpoints. Midpoint of AC = (−0.5, −0.5) and midpoint of BD = (−0.5, −0.5). They are the same. ✓
Watch out. Keep the corners in order round the shape: A, B, C, D. If you join them in the wrong order you get a bow-tie, not a parallelogram.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. In a parallelogram, opposite sides are parallel and equal, so the move from B to A is the same as the move from C to D.
- Plot the three points A, B and C. A sketch helps you see where the fourth corner has to go. The corners are labelled in order round the shape.A = (−3, −1), B = (2, −3), C = (6, −3)
- Work out the move from B to A. In a parallelogram, opposite sides are parallel and equal in length. So side CD is the same move as side BA.across: −3 − 2 = −5 (5 left)
up/down: −1 − (−3) = 2 (2 up) - Start at C and make the same move to reach D. C is at (6, −3). Add the same across and up/down numbers.x: 6 + (−5) = 1
y: −3 + 2 = −1 - Write the coordinates and draw the shape. D = (1, −1)
Check. The diagonals of a parallelogram cross at their midpoints. Midpoint of AC = (1.5, −2) and midpoint of BD = (1.5, −2). They are the same. ✓
Watch out. Keep the corners in order round the shape: A, B, C, D. If you join them in the wrong order you get a bow-tie, not a parallelogram.
Method. Find the base and height from the coordinates, then area of a triangle = ½ × base × height.
- Sketch the triangle. Plot the three points and join them.
- Find the base and the height. Two sides meet at a right angle at B, one horizontal and one vertical. These are the base and the height. Subtract coordinates to find each length.base = 6, height = 4
- Use area = ½ × base × height. The triangle is half of a rectangle that is base by height.6 × 4 ÷ 2 = 24 ÷ 2 = 12
Check. The rectangle round the triangle has area 6 × 4 = 24; half of it is 12. ✓
Watch out. Use the perpendicular height, not a slanted side.
Watch out
Order matters. (3, 2) and (2, 3) are different points. Also take care with minus signs: (−2, 5) is left of the y-axis, so it is in Quadrant 2, not Quadrant 4. When you count the distance between points, use bigger − smaller, and remember 5 − (−3) = 8.Exercise 12A Coordinates and shapes on a grid
Non-calculatorKey words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. Read a point by going across from the origin first (the x-coordinate), then up or down (the y-coordinate).
- Start at the origin (0, 0), where the two axes cross. Every coordinate is measured from the origin.
- Find the x-coordinate: how far across to reach A? Move along the horizontal axis first. Right is positive and left is negative.Go 3 squares to the right, so x = 3.
- Find the y-coordinate: how far up or down to reach A? Then move parallel to the vertical axis. Up is positive and down is negative.Go 4 squares up, so y = 4.
- Write the coordinates as (x, y), across first. The order matters: (3, 2) is not the same point as (2, 3).A = (3, 4)
Check. Start at (0, 0), go across 3 and then up 4: you arrive at A. ✓
Watch out. Across first, then up or down. Many mistakes come from writing the numbers the wrong way round.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. Read a point by going across from the origin first (the x-coordinate), then up or down (the y-coordinate).
- Start at the origin (0, 0), where the two axes cross. Every coordinate is measured from the origin.
- Find the x-coordinate: how far across to reach B? Move along the horizontal axis first. Right is positive and left is negative.Go 2 squares to the left, so x = −2.
- Find the y-coordinate: how far up or down to reach B? Then move parallel to the vertical axis. Up is positive and down is negative.Go 5 squares up, so y = 5.
- Write the coordinates as (x, y), across first. The order matters: (3, 2) is not the same point as (2, 3).B = (−2, 5)
Check. Start at (0, 0), go across −2 and then up 5: you arrive at B. ✓
Watch out. Across first, then up or down. Many mistakes come from writing the numbers the wrong way round.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. Read a point by going across from the origin first (the x-coordinate), then up or down (the y-coordinate).
- Start at the origin (0, 0), where the two axes cross. Every coordinate is measured from the origin.
- Find the x-coordinate: how far across to reach C? Move along the horizontal axis first. Right is positive and left is negative.Go 5 squares to the left, so x = −5.
- Find the y-coordinate: how far up or down to reach C? Then move parallel to the vertical axis. Up is positive and down is negative.Go 2 squares up, so y = 2.
- Write the coordinates as (x, y), across first. The order matters: (3, 2) is not the same point as (2, 3).C = (−5, 2)
Check. Start at (0, 0), go across −5 and then up 2: you arrive at C. ✓
Watch out. Across first, then up or down. Many mistakes come from writing the numbers the wrong way round.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. Read a point by going across from the origin first (the x-coordinate), then up or down (the y-coordinate).
- Start at the origin (0, 0), where the two axes cross. Every coordinate is measured from the origin.
- Find the x-coordinate: how far across to reach D? Move along the horizontal axis first. Right is positive and left is negative.Go 4 squares to the left, so x = −4.
- Find the y-coordinate: how far up or down to reach D? Then move parallel to the vertical axis. Up is positive and down is negative.Go 3 squares down, so y = −3.
- Write the coordinates as (x, y), across first. The order matters: (3, 2) is not the same point as (2, 3).D = (−4, −3)
Check. Start at (0, 0), go across −4 and then down 3: you arrive at D. ✓
Watch out. Across first, then up or down. Many mistakes come from writing the numbers the wrong way round.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. Read a point by going across from the origin first (the x-coordinate), then up or down (the y-coordinate).
- Start at the origin (0, 0), where the two axes cross. Every coordinate is measured from the origin.
- Find the x-coordinate: how far across to reach E? Move along the horizontal axis first. Right is positive and left is negative.Go 2 squares to the right, so x = 2.
- Find the y-coordinate: how far up or down to reach E? Then move parallel to the vertical axis. Up is positive and down is negative.Go 4 squares down, so y = −4.
- Write the coordinates as (x, y), across first. The order matters: (3, 2) is not the same point as (2, 3).E = (2, −4)
Check. Start at (0, 0), go across 2 and then down 4: you arrive at E. ✓
Watch out. Across first, then up or down. Many mistakes come from writing the numbers the wrong way round.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. Read a point by going across from the origin first (the x-coordinate), then up or down (the y-coordinate).
- Start at the origin (0, 0), where the two axes cross. Every coordinate is measured from the origin.
- Find the x-coordinate: how far across to reach F? Move along the horizontal axis first. Right is positive and left is negative.Go 5 squares to the right, so x = 5.
- Find the y-coordinate: how far up or down to reach F? Then move parallel to the vertical axis. Up is positive and down is negative.Go 1 square down, so y = −1.
- Write the coordinates as (x, y), across first. The order matters: (3, 2) is not the same point as (2, 3).F = (5, −1)
Check. Start at (0, 0), go across 5 and then down 1: you arrive at F. ✓
Watch out. Across first, then up or down. Many mistakes come from writing the numbers the wrong way round.
Method. The point where the axes cross has a special name.
- Recall what (0, 0) means. Both coordinates are zero, so it is neither right nor up.
- Write the word. It is where the two axes meet.origin
Check. The origin is where you start every time you read coordinates. ✓
Method. Remember: x goes along the corridor.
- Think about the order. The first coordinate (x) tells you how far across to go.
- Write the letter. Across is horizontal.x-axis
Check. In (3, 5) the 3 is the x value and it moves you across. ✓
Method. The y-coordinate is the second number.
- Split the pair. (x, y) is written with x first.x = −3, y = 5
- Choose the second. y = 5
Check. 5 means 5 units up from the x-axis. ✓
Method. A point with y = 0 is not up or down at all.
- Look at the y-coordinate. It is 0, so the point is level with the origin.y = 0
- Decide the axis. Points with y = 0 lie on the horizontal axis.x-axis
Check. From the origin, you only move 4 across and not up or down. ✓
Key words. The two axes cut the grid into four quadrants. They are numbered 1, 2, 3, 4 going anticlockwise from the top right.
Method. The signs of the two coordinates tell you which quadrant (or axis) the point is in.
- Look at the sign of the x-coordinate. The x-coordinate tells us whether the point is left or right of the vertical axis.x = 3 is positive (right of the y-axis).
- Look at the sign of the y-coordinate. The y-coordinate tells us whether the point is above or below the horizontal axis.y = 5 is positive (above the x-axis).
- Match the signs to a quadrant. Quadrant 1 is (+, +), Quadrant 2 is (−, +), Quadrant 3 is (−, −), Quadrant 4 is (+, −). If a coordinate is 0 the point is on an axis, not in a quadrant.(3, 5) is Quadrant 1.
Check. Picture the point: right and above the origin. ✓
Watch out. The quadrants are numbered anticlockwise starting at the top right. Quadrant 2 is top left, not bottom right.
Key words. The two axes cut the grid into four quadrants. They are numbered 1, 2, 3, 4 going anticlockwise from the top right.
Method. The signs of the two coordinates tell you which quadrant (or axis) the point is in.
- Look at the sign of the x-coordinate. The x-coordinate tells us whether the point is left or right of the vertical axis.x = −2 is negative (left of the y-axis).
- Look at the sign of the y-coordinate. The y-coordinate tells us whether the point is above or below the horizontal axis.y = 4 is positive (above the x-axis).
- Match the signs to a quadrant. Quadrant 1 is (+, +), Quadrant 2 is (−, +), Quadrant 3 is (−, −), Quadrant 4 is (+, −). If a coordinate is 0 the point is on an axis, not in a quadrant.(−2, 4) is Quadrant 2.
Check. Picture the point: left and above the origin. ✓
Watch out. The quadrants are numbered anticlockwise starting at the top right. Quadrant 2 is top left, not bottom right.
Key words. The two axes cut the grid into four quadrants. They are numbered 1, 2, 3, 4 going anticlockwise from the top right.
Method. The signs of the two coordinates tell you which quadrant (or axis) the point is in.
- Look at the sign of the x-coordinate. The x-coordinate tells us whether the point is left or right of the vertical axis.x = −5 is negative (left of the y-axis).
- Look at the sign of the y-coordinate. The y-coordinate tells us whether the point is above or below the horizontal axis.y = −1 is negative (below the x-axis).
- Match the signs to a quadrant. Quadrant 1 is (+, +), Quadrant 2 is (−, +), Quadrant 3 is (−, −), Quadrant 4 is (+, −). If a coordinate is 0 the point is on an axis, not in a quadrant.(−5, −1) is Quadrant 3.
Check. Picture the point: left and below the origin. ✓
Watch out. The quadrants are numbered anticlockwise starting at the top right. Quadrant 2 is top left, not bottom right.
Key words. The two axes cut the grid into four quadrants. They are numbered 1, 2, 3, 4 going anticlockwise from the top right.
Method. The signs of the two coordinates tell you which quadrant (or axis) the point is in.
- Look at the sign of the x-coordinate. The x-coordinate tells us whether the point is left or right of the vertical axis.x = 6 is positive (right of the y-axis).
- Look at the sign of the y-coordinate. The y-coordinate tells us whether the point is above or below the horizontal axis.y = −3 is negative (below the x-axis).
- Match the signs to a quadrant. Quadrant 1 is (+, +), Quadrant 2 is (−, +), Quadrant 3 is (−, −), Quadrant 4 is (+, −). If a coordinate is 0 the point is on an axis, not in a quadrant.(6, −3) is Quadrant 4.
Check. Picture the point: right and below the origin. ✓
Watch out. The quadrants are numbered anticlockwise starting at the top right. Quadrant 2 is top left, not bottom right.
Key words. The two axes cut the grid into four quadrants. They are numbered 1, 2, 3, 4 going anticlockwise from the top right.
Method. The signs of the two coordinates tell you which quadrant (or axis) the point is in.
- Look at the sign of the x-coordinate. The x-coordinate tells us whether the point is left or right of the vertical axis.x = 0 is zero (on the y-axis).
- Look at the sign of the y-coordinate. The y-coordinate tells us whether the point is above or below the horizontal axis.y = −4 is negative (below the x-axis).
- Match the signs to a quadrant. Quadrant 1 is (+, +), Quadrant 2 is (−, +), Quadrant 3 is (−, −), Quadrant 4 is (+, −). If a coordinate is 0 the point is on an axis, not in a quadrant.(0, −4) is on the y-axis.
Check. Picture the point: straight up or down and below the origin. ✓
Watch out. The quadrants are numbered anticlockwise starting at the top right. Quadrant 2 is top left, not bottom right.
Key words. The two axes cut the grid into four quadrants. They are numbered 1, 2, 3, 4 going anticlockwise from the top right.
Method. The signs of the two coordinates tell you which quadrant (or axis) the point is in.
- Look at the sign of the x-coordinate. The x-coordinate tells us whether the point is left or right of the vertical axis.x = −7 is negative (left of the y-axis).
- Look at the sign of the y-coordinate. The y-coordinate tells us whether the point is above or below the horizontal axis.y = 0 is zero (on the x-axis).
- Match the signs to a quadrant. Quadrant 1 is (+, +), Quadrant 2 is (−, +), Quadrant 3 is (−, −), Quadrant 4 is (+, −). If a coordinate is 0 the point is on an axis, not in a quadrant.(−7, 0) is on the x-axis.
Check. Picture the point: left and level with the origin. ✓
Watch out. The quadrants are numbered anticlockwise starting at the top right. Quadrant 2 is top left, not bottom right.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. Plot each point by going across first, then up or down. Mark it with a small cross.
- Plot A (−3, −1). Start at the origin. Move left 3 and then down 1.A: across −3, down 1
- Plot B (3, −1). Start at the origin. Move right 3 and then down 1.B: across 3, down 1
- Plot C (3, 2). Start at the origin. Move right 3 and then up 2.C: across 3, up 2
- Plot D (−3, 2). Start at the origin. Move left 3 and then up 2.D: across −3, up 2
- Join the points in order with a ruler. This shows the shape. The shape has four right angles and opposite sides equal: it is a rectangle, 6 units by 3 units.
Check. Re-read each plotted point from the grid and compare with the coordinates in the question. ✓
Watch out. Across first, then up or down. Check negative numbers go left or down.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. In a rectangle, opposite sides are parallel and equal, so the move from B to A is the same as the move from C to D.
- Plot the three points A, B and C. A sketch helps you see where the fourth corner has to go. The corners are labelled in order round the shape.A = (−4, −2), B = (3, −2), C = (3, 1)
- Work out the move from B to A. In a rectangle, opposite sides are parallel and equal in length, and every angle is 90°. So side CD is the same move as side BA.across: −4 − 3 = −7 (7 left)
up/down: −2 − (−2) = 0 (0) - Start at C and make the same move to reach D. C is at (3, 1). Add the same across and up/down numbers.x: 3 + (−7) = −4
y: 1 + 0 = 1 - Write the coordinates and draw the shape. D = (−4, 1)
Check. The diagonals of a rectangle cross at their midpoints. Midpoint of AC = (−0.5, −0.5) and midpoint of BD = (−0.5, −0.5). They are the same. ✓
Watch out. Keep the corners in order round the shape: A, B, C, D. If you join them in the wrong order you get a bow-tie, not a parallelogram.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. In a rectangle, opposite sides are parallel and equal, so the move from B to A is the same as the move from C to D.
- Plot the three points A, B and C. A sketch helps you see where the fourth corner has to go. The corners are labelled in order round the shape.A = (−3, 1), B = (−3, −3), C = (2, −3)
- Work out the move from B to A. In a rectangle, opposite sides are parallel and equal in length, and every angle is 90°. So side CD is the same move as side BA.across: −3 − (−3) = 0 (0)
up/down: 1 − (−3) = 4 (4 up) - Start at C and make the same move to reach D. C is at (2, −3). Add the same across and up/down numbers.x: 2 + 0 = 2
y: −3 + 4 = 1 - Write the coordinates and draw the shape. D = (2, 1)
Check. The diagonals of a rectangle cross at their midpoints. Midpoint of AC = (−0.5, −1) and midpoint of BD = (−0.5, −1). They are the same. ✓
Watch out. Keep the corners in order round the shape: A, B, C, D. If you join them in the wrong order you get a bow-tie, not a parallelogram.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. In a rectangle, opposite sides are parallel and equal, so the move from B to A is the same as the move from C to D.
- Plot the three points A, B and C. A sketch helps you see where the fourth corner has to go. The corners are labelled in order round the shape.A = (−5, 4), B = (1, 4), C = (1, −1)
- Work out the move from B to A. In a rectangle, opposite sides are parallel and equal in length, and every angle is 90°. So side CD is the same move as side BA.across: −5 − 1 = −6 (6 left)
up/down: 4 − 4 = 0 (0) - Start at C and make the same move to reach D. C is at (1, −1). Add the same across and up/down numbers.x: 1 + (−6) = −5
y: −1 + 0 = −1 - Write the coordinates and draw the shape. D = (−5, −1)
Check. The diagonals of a rectangle cross at their midpoints. Midpoint of AC = (−2, 1.5) and midpoint of BD = (−2, 1.5). They are the same. ✓
Watch out. Keep the corners in order round the shape: A, B, C, D. If you join them in the wrong order you get a bow-tie, not a parallelogram.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. In a parallelogram, opposite sides are parallel and equal, so the move from B to A is the same as the move from C to D.
- Plot the three points A, B and C. A sketch helps you see where the fourth corner has to go. The corners are labelled in order round the shape.A = (−2, −1), B = (1, −3), C = (5, −3)
- Work out the move from B to A. In a parallelogram, opposite sides are parallel and equal in length. So side CD is the same move as side BA.across: −2 − 1 = −3 (3 left)
up/down: −1 − (−3) = 2 (2 up) - Start at C and make the same move to reach D. C is at (5, −3). Add the same across and up/down numbers.x: 5 + (−3) = 2
y: −3 + 2 = −1 - Write the coordinates and draw the shape. D = (2, −1)
Check. The diagonals of a parallelogram cross at their midpoints. Midpoint of AC = (1.5, −2) and midpoint of BD = (1.5, −2). They are the same. ✓
Watch out. Keep the corners in order round the shape: A, B, C, D. If you join them in the wrong order you get a bow-tie, not a parallelogram.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. In a parallelogram, opposite sides are parallel and equal, so the move from B to A is the same as the move from C to D.
- Plot the three points A, B and C. A sketch helps you see where the fourth corner has to go. The corners are labelled in order round the shape.A = (0, 2), B = (−3, −1), C = (1, −1)
- Work out the move from B to A. In a parallelogram, opposite sides are parallel and equal in length. So side CD is the same move as side BA.across: 0 − (−3) = 3 (3 right)
up/down: 2 − (−1) = 3 (3 up) - Start at C and make the same move to reach D. C is at (1, −1). Add the same across and up/down numbers.x: 1 + 3 = 4
y: −1 + 3 = 2 - Write the coordinates and draw the shape. D = (4, 2)
Check. The diagonals of a parallelogram cross at their midpoints. Midpoint of AC = (0.5, 0.5) and midpoint of BD = (0.5, 0.5). They are the same. ✓
Watch out. Keep the corners in order round the shape: A, B, C, D. If you join them in the wrong order you get a bow-tie, not a parallelogram.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. In a parallelogram, opposite sides are parallel and equal, so the move from B to A is the same as the move from C to D.
- Plot the three points A, B and C. A sketch helps you see where the fourth corner has to go. The corners are labelled in order round the shape.A = (−4, 3), B = (−2, −2), C = (3, −2)
- Work out the move from B to A. In a parallelogram, opposite sides are parallel and equal in length. So side CD is the same move as side BA.across: −4 − (−2) = −2 (2 left)
up/down: 3 − (−2) = 5 (5 up) - Start at C and make the same move to reach D. C is at (3, −2). Add the same across and up/down numbers.x: 3 + (−2) = 1
y: −2 + 5 = 3 - Write the coordinates and draw the shape. D = (1, 3)
Check. The diagonals of a parallelogram cross at their midpoints. Midpoint of AC = (−0.5, 0.5) and midpoint of BD = (−0.5, 0.5). They are the same. ✓
Watch out. Keep the corners in order round the shape: A, B, C, D. If you join them in the wrong order you get a bow-tie, not a parallelogram.
Method. Left/down are negative; right/up are positive. Write across first.
- Find the x-coordinate. Left of the y-axis means negative x. It is 4 units away.x = −4
- Find the y-coordinate. Below the x-axis means negative y. It is 3 units away.y = −3
- Write (x, y). P = (−4, −3)
Check. P is in Quadrant 3, which has (−, −) coordinates. ✓
Method. Right/up are positive. Write across first.
- Find the x-coordinate. Right of the y-axis means positive x.x = 5
- Find the y-coordinate. Above the x-axis means positive y.y = 2
- Write (x, y). Q = (5, 2)
Check. Q is in Quadrant 1, which has (+, +) coordinates. ✓
Method. A point on the x-axis has y = 0.
- Find the x-coordinate. Left of the origin means negative.x = −6
- Find the y-coordinate. Any point on the x-axis has y = 0.y = 0
- Write (x, y). R = (−6, 0)
Check. R is level with the origin, so it has no up or down. ✓
Method. A point on the y-axis has x = 0.
- Find the x-coordinate. Any point on the y-axis has x = 0.x = 0
- Find the y-coordinate. Below the origin is negative.y = −5
- Write (x, y). S = (0, −5)
Check. S is directly below the origin. ✓
Method. When two points have the same x- or y-coordinate, the distance between them is the difference of the other coordinate.
- Look at which coordinate is the same. If one coordinate matches, the line is parallel to an axis. Here both y-coordinates are 2, so the line is horizontal.(−3, 2) and (5, 2)
- Use the x-coordinates, which are different. The distance is the bigger number minus the smaller number. This works even with negatives.5 − (−3) = 8
Check. Count it on the grid: from −3 to 0 is 3 and from 0 to 5 is 5, so 3 + 5 = 8. ✓
Watch out. Subtracting a negative: 5 − (−3) = 5 + 3 = 8. It is not 2.
Method. When two points have the same x- or y-coordinate, the distance between them is the difference of the other coordinate.
- Look at which coordinate is the same. If one coordinate matches, the line is parallel to an axis. Here both x-coordinates are 4, so the line is vertical.(4, −1) and (4, 6)
- Use the y-coordinates, which are different. The distance is the bigger number minus the smaller number. This works even with negatives.6 − (−1) = 7
Check. Count it on the grid: from −1 to 0 is 1 and from 0 to 6 is 6, so 1 + 6 = 7. ✓
Watch out. Subtracting a negative: 5 − (−3) = 5 + 3 = 8. It is not 2.
Method. When two points have the same x- or y-coordinate, the distance between them is the difference of the other coordinate.
- Look at which coordinate is the same. If one coordinate matches, the line is parallel to an axis. Here both x-coordinates are −6, so the line is vertical.(−6, −2) and (−6, −7)
- Use the y-coordinates, which are different. The distance is the bigger number minus the smaller number. This works even with negatives.−2 − (−7) = 5
Check. Count the squares along the line between the two points: 5 units. ✓
Watch out. Subtracting a negative: 5 − (−3) = 5 + 3 = 8. It is not 2.
Method. When two points have the same x- or y-coordinate, the distance between them is the difference of the other coordinate.
- Look at which coordinate is the same. If one coordinate matches, the line is parallel to an axis. Here both y-coordinates are 3, so the line is horizontal.(−2, 3) and (7, 3)
- Use the x-coordinates, which are different. The distance is the bigger number minus the smaller number. This works even with negatives.7 − (−2) = 9
Check. Count it on the grid: from −2 to 0 is 2 and from 0 to 7 is 7, so 2 + 7 = 9. ✓
Watch out. Subtracting a negative: 5 − (−3) = 5 + 3 = 8. It is not 2.
Key words. A coordinate pair such as (3, 2) gives the position of a point. The first number is the x-coordinate (how far across, along the horizontal axis); the second is the y-coordinate (how far up or down). The origin is the point (0, 0) where the axes cross.
Method. Follow each move in turn, changing only x for left/right moves and only y for up/down moves.
- Start at the given point. We keep track of the x- and y-coordinates after every move.start = (−4, 3)
- Move right 6. Moving right adds to the x-coordinate.x: −4 + 6 = 2
now at (2, 3) - Move down 5. Moving down subtracts from the y-coordinate.y: 3 − 5 = −2
now at (2, −2) - Move left 3. Moving left subtracts from the x-coordinate.x: 2 − 3 = −1
now at (−1, −2) - Move up 8. Moving up adds to the y-coordinate.y: −2 + 8 = 6
now at (−1, 6)
Check. Total across: 3; total up/down: 3. Start + totals = (−1, 6). ✓
Watch out. Left and down make the numbers smaller (go towards negative). Right and up make them bigger.
Method. Find the length of each side by subtracting coordinates, then use perimeter = 2(width + height) and area = width × height.
- Draw a sketch. The sketch shows the rectangle and which sides are horizontal and vertical.
- Find the length of the horizontal sides. The two x-coordinates are different; subtract them.4 − (−2) = 6
- Find the length of the vertical sides. The two y-coordinates are different; subtract them.3 − (−1) = 4
- Calculate the perimeter. Perimeter is the distance all the way round: two widths and two heights.2 × 6 + 2 × 4 = 12 + 8 = 20
- Calculate the area. Area of a rectangle = width × height.6 × 4 = 24
Check. Count squares on the grid: the rectangle is 6 squares wide and 4 squares high, so it covers 6 × 4 = 24 squares. ✓
Watch out. Perimeter and area are different: perimeter is a length (add the sides), area is in square units (multiply).
Method. Find the base and height from the coordinates, then area of a triangle = ½ × base × height.
- Sketch the triangle. Plot the three points and join them.
- Find the base and the height. Two sides meet at a right angle at B, one horizontal and one vertical. These are the base and the height. Subtract coordinates to find each length.base = 6, height = 4
- Use area = ½ × base × height. The triangle is half of a rectangle that is base by height.6 × 4 ÷ 2 = 24 ÷ 2 = 12
Check. The rectangle round the triangle has area 6 × 4 = 24; half of it is 12. ✓
Watch out. Use the perpendicular height, not a slanted side.
Method. Find the base and height from the coordinates, then area of a triangle = ½ × base × height.
- Sketch the triangle. Plot the three points and join them.
- Find the base and the height. Two sides meet at a right angle at A, one horizontal and one vertical. These are the base and the height. Subtract coordinates to find each length.base = 5, height = 6
- Use area = ½ × base × height. The triangle is half of a rectangle that is base by height.5 × 6 ÷ 2 = 30 ÷ 2 = 15
Check. The rectangle round the triangle has area 5 × 6 = 30; half of it is 15. ✓
Watch out. Use the perpendicular height, not a slanted side.
Method. Check the signs of the coordinates against the quadrants.
- Look at the signs. −2 is negative (left of the y-axis). 5 is positive (above the x-axis).(−, +)
- Match to a quadrant. Quadrant 4 is bottom right, with signs (+, −). Quadrant 2 is top left, with signs (−, +).(−, +) is Quadrant 2
- Write the conclusion. Say what is wrong, not just “no”.Sam is wrong. (−2, 5) is 2 left and 5 up, so it is in Quadrant 2.
Check. Sketch it: 2 left and 5 up is in the top left corner, which is Quadrant 2. ✓
Watch out. Quadrant 4 is the one at the bottom right, not the top left.
Method. Show that two sides meet at a right angle. A horizontal line and a vertical line always meet at 90°.
- Look at AB. A(1, 3) and B(4, 3) have the same y-coordinate, so AB is a horizontal line.AB is horizontal, length 4 − 1 = 3
- Look at BC. B(4, 3) and C(4, −2) have the same x-coordinate, so BC is a vertical line.BC is vertical, length 3 − (−2) = 5
- Conclude. A horizontal line and a vertical line meet at a right angle.Angle ABC = 90°, so the triangle is right-angled at B. Its area is ½ × 3 × 5 = 7.5.
- Check with a sketch. The picture shows the right angle at B.
Check. Plot the three points: the corner at B looks like the corner of a sheet of paper. ✓
Watch out. In a “show that” question, finish with a sentence that states the conclusion.
Method. The diagonal joins opposite corners. In a square the other two corners are the other ends of the horizontal and vertical lines through these corners.
- Plot A and C. A is bottom-left and C is top-right.A(−2, −2), C(2, 2)
- Find B and D. B shares the x-coordinate of C and the y-coordinate of A. D shares the x-coordinate of A and the y-coordinate of C.B = (2, −2), D = (−2, 2)
- Find the side and the area. AB goes from x = −2 to x = 2.side = 2 − (−2) = 4; area = 4 × 4 = 16
- Check with a sketch. The picture shows a square.
Check. All four sides are 4 units long and the angles are right angles, so it is a square. ✓
Watch out. A square has four equal sides. If you get different side lengths, a vertex is wrong.
Method. East is to the right (+) and south is down (−).
- Start at the school (−3, 2). East means move right, so add to x.x: −3 + 5 = 2
- South means move down, so subtract from y. y: 2 − 4 = −2
Check. From (2, −2) go 5 left and 4 up to return to (−3, 2). ✓
Method. Directly north of the park means the same x-coordinate. Level with the school means the same y-coordinate as the school.
- Same x as the park. Directly north means straight up.x = 2
- Same y as the school. Level with means the same height.y = 2
Check. (2, 2) is directly above (2, −2) and level with (−3, 2). ✓
Method. The school and library have the same y-coordinate, so the line is horizontal; subtract the x-coordinates.
- Check the line is horizontal. Both have y = 2.School (−3, 2), library (2, 2)
- Subtract. Bigger − smaller.2 − (−3) = 5 km
Check. It is the same as the park being 5 km east of the school. ✓
Method. Use the fact that every point on the y-axis has x = 0 and every point on the x-axis has y = 0.
- Find a. P is on the y-axis, so its x-coordinate is 0.a = 0
- Find b. Q is on the x-axis, so its y-coordinate is 0.b = 0
- Write the points. Put the numbers back into the coordinates.P = (0, 3) and Q = (−2, 0)
Check. P is 3 up from the origin; Q is 2 left of the origin. ✓
Watch out. The y-axis is the vertical line where x = 0.