The Number Plane
Every question with a fully worked solution.
- The Number Plane - Video - Identify the Quadrant of a Point on the Coordinate Plane Watch
Theory
Every point on the number plane has coordinates \((x, y)\): \(x\) is how far right or left of the origin, \(y\) how far up or down. The axes cut the plane into four quadrants, and the signs of \(x\) and \(y\) tell you which one: \((-1, 3)\) is in the 2nd. This Year 8 maths page shows how.
Reading and plotting. Start at the origin \(O(0, 0)\). Go across first: positive \(x\) is right, negative \(x\) is left. Then positive \(y\) is up, negative \(y\) is down.
Quadrants are numbered anticlockwise from the top right. For a point \(P(x, y)\): \(x > 0, y > 0\) is the 1st (I); \(x < 0, y > 0\) the 2nd (II); \(x < 0, y < 0\) the 3rd (III); \(x > 0, y < 0\) the 4th (IV). Only the signs decide, so \((-9.8, 8)\) is in the 2nd.
On an axis. A point on an axis is in no quadrant: on the \(x\)-axis \(y = 0\), on the \(y\)-axis \(x = 0\).
Between grid lines. A point halfway between two grid lines has a coordinate ending in \(.5\), such as \(-1.5\). The distance from the \(y\)-axis is the size of \(x\); from the \(x\)-axis, the size of \(y\).
Lengths. Two points with the same \(y\) lie on a horizontal line; two with the same \(x\) lie on a vertical line. The length is the larger coordinate minus the smaller, even across an axis: \(4 - (-3) = 7\) units.
Shapes. Once the side lengths are known, use the area formulas, such as \(A = \dfrac{1}{2}(a + b)h\) for a trapezium and \(A = \dfrac{1}{2}xy\) for a kite or rhombus with diagonals \(x\) and \(y\). Areas are in square units. Where a horizontal and a vertical path meet at a right angle, Pythagoras’ theorem gives the straight-line distance between their far ends.
The signs in each quadrant:
| Quadrant | \(x\) | \(y\) | Example |
|---|---|---|---|
| 1st (I) | + | + | \((2, 5)\) |
| 2nd (II) | − | + | \((-1, 3)\) |
| 3rd (III) | − | − | \((-1, -2)\) |
| 4th (IV) | + | − | \((6, -3)\) |
Plotting a point (x, y)
- Start at the origin.
- Move across \(x\) units: right if positive, left if negative.
- Move up or down \(y\) units: up if positive, down if negative.
- Mark the point and label it.
Finding a horizontal or vertical length
- Check which coordinate is the same (same \(y\): horizontal; same \(x\): vertical).
- Subtract the smaller other coordinate from the larger one.
(i) \(P(-1, 3)\)
(ii) \(Q(2, 5)\)
(i) \(x < 0\) and \(y > 0\), so the 2nd quadrant (II).
(ii) \(x > 0\) and \(y > 0\), so the 1st quadrant (I).
(i) \(L(-1, -2)\)
(ii) \(M(6, -3)\)
(i) \(x < 0\) and \(y < 0\), so the 3rd quadrant (III).
(ii) \(x > 0\) and \(y < 0\), so the 4th quadrant (IV).
\(P(-4, 2.5)\), 2nd quadrant
\(Q(3, -4)\), 4th quadrant
\(R(-2, 0)\) and \(S(0, 4)\) are on the axes, so they are in no quadrant.
(i) Give the coordinates of \(V\).
(ii) \(W\) is \(6\) units right of \(V\). Give \(W\) and its quadrant.
In the 3rd quadrant both coordinates are negative. Moving right adds to \(x\) and leaves \(y\) alone.
(i) \(V(-4, -1.5)\)
(ii) \(-4 + 6 = 2\), so \(W(2, -1.5)\), in the 4th quadrant.
\(AB\) and \(DC\) are both horizontal, so they are the parallel sides of a trapezium.
| \(AB\) | \(=\) | \(2 - (-5) = 7\) |
| \(DC\) | \(=\) | \(1 - (-3) = 4\) |
| \(h\) | \(=\) | \(3 - (-1) = 4\) |
| \(A\) | \(=\) | \(\dfrac{1}{2} \times (7 + 4) \times 4 = 22\) |
The bed has an area of \(22\) m\(^2\).
(ii) On a map with \(1\) unit \(= 1\) km, Sam walks from \((-4, 3)\) to \((5, 3)\), then to \((5, -9)\), then straight back. How far does he walk?
(i) The diagonals are \(5 - (-3) = 8\) and \(2 - (-2) = 4\).
(ii) The legs are \(5 - (-4) = 9\) km and \(3 - (-9) = 12\) km, at right angles.
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 8 \times 4 = 16\) |
| \(\text{Back}\) | \(=\) | \(\sqrt{9^2 + 12^2} = 15\) |
| \(\text{Total}\) | \(=\) | \(9 + 12 + 15 = 36\) |
(i) \(16\) square units
(ii) \(36\) km
(i) What are the signs of \(p\) and \(q\)?
(ii) In which quadrant is \((q, p)\)?
(iii) In which quadrant is \((p, -q)\)?
(i) \(p\) is negative and \(q\) is positive.
(ii) \((q, p)\) is \((+, -)\): the 4th quadrant.
(iii) \((p, -q)\) is \((-, -)\): the 3rd quadrant. Test: \(M(-3, 5)\) gives \((5, -3)\) and \((-3, -5)\).
Common pitfalls
Frequently asked questions
What are coordinates?
A pair of numbers (x, y) that give the position of a point: x across from the origin, then y up or down.
How are the quadrants numbered?
Anticlockwise from the top right: 1st top right, 2nd top left, 3rd bottom left, 4th bottom right.
Which quadrant is a point with a negative x and a positive y in?
The 2nd quadrant.
Which quadrant is the point (0, 3) in?
None. It is on the y-axis.
How do I find the distance between (−4, 2) and (3, 2)?
They have the same y, so the line is horizontal. Subtract the x-values: 3 − (−4) = 7 units.