Coordinate geometry

Q: What are the four quadrants?

The axes split the plane into four parts: I (+, +), II (-, +), III (-, -) and IV (+, -), counted anticlockwise from the top right.

Q: What are the coordinates of a point on the x-axis?

Its y-coordinate is 0, for example (5, 0). A point on the y-axis has x-coordinate 0.

Coordinate geometry describes points by two numbers $(x, y)$ : how far across and how far up from the origin. Once points have coordinates, lengths, midpoints and areas can be worked out with formulas instead of by drawing.

Formulas

Distance between two points

A B = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Midpoint, and the point dividing AB in the ratio m : n

(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}) (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})

Area of a triangle from its vertices

\frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

Worked examples

Example 1. Find the distance between

(1, 2)

and

(4, 6)

$(4 - 1)^{2} + (6 - 2)^{2} = 9 + 16 = 25$ .

Answer: 5 units

Example 2. Find the midpoint of

(- 2, 3)

and

(6, 7)

$(\frac{- 2 + 6}{2}, \frac{3 + 7}{2}) = (2, 5)$ .

Answer:

(2, 5)

Example 3. Find the point that divides

A (2, 1)

and

B (8, 7)

in the ratio

1 : 2

$m = 1, n = 2 : x = \frac{1 \times 8 + 2 \times 2}{3} = \frac{12}{3} = 4$ .
$y = \frac{1 \times 7 + 2 \times 1}{3} = \frac{9}{3} = 3$ .

Answer:

(4, 3)

Example 4. Find the area of the triangle with vertices

(0, 0), (4, 0)

and

(0, 3)

$\frac{1}{2} ∣ 0 (0 - 3) + 4 (3 - 0) + 0 (0 - 0) ∣ = \frac{1}{2} \times 12$ .
Check: it is a right-angled triangle with legs 4 and 3, so $\frac{1}{2} \times 4 \times 3 = 6$ . ✓

Answer: 6 square units

Common mistakes

Writing $(y, x)$ instead of $(x, y)$ . The across-value always comes first.
Losing a minus sign when squaring a difference: $(- 3)^{2} = 9$ , never $- 9$ .
Swapping $m$ and $n$ in the section formula. $m$ goes with the second point $B$ .

Practice questions with answers

1. What is the abscissa

(x

-coordinate) of the point (−7, 4)?

A−7
B4
C7
D−4

Show answer

Answer: A) −7

Abscissa = first coordinate =

x = - 7

2. The point where the

x

-axis and

y

-axis intersect is called the ____.

AOrigin
BVertex
CCentre
DNode

Show answer

Answer: A) Origin

The two axes meet at the origin, whose coordinates are (0, 0).

3. In which quadrant does the point (3, 5) lie?

AQuadrant I
BQuadrant II
CQuadrant III
DQuadrant IV

Show answer

Answer: A) Quadrant I

Both coordinates are positive (+, +), so the point lies in Quadrant I.

4. In the ordered pair (5, 3), what is the

x

-coordinate (abscissa)?

Show answer

Answer: A) 5

(x, y)

the first number is the

x

-coordinate (abscissa). Here it is 5.

5. Distance between (0, 0) and (3, 4).

A5
B7
C25
D3.5

Show answer

Answer: A) 5

Distance =

3^{2} + 4^{2} = 9 + 16 = 25 = 5

6. What is the distance formula between points A(x1, y1) and B(x2, y2)?

A $(x 2 - x 1)^{2} + (y 2 - y 1)^{2}$
B $(x 2 - x 1)^{2} + (y 2 - y 1)^{2}$
C $(x 2 + x 1)^{2} + (y 2 + y 1)^{2}$
D|x2 - x1| + |y2 - y1|

Show answer

Answer: A)

(x 2 - x 1)^{2} + (y 2 - y 1)^{2}

The distance formula is

(x 2 - x 1)^{2} + (y 2 - y 1)^{2}

, derived from the Pythagoras theorem.

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Frequently asked questions

What are the four quadrants?

The axes split the plane into four parts: I $(+, +)$ , II $(-, +)$ , III $(-, -)$ and IV $(+, -)$ , counted anticlockwise from the top right.

What are the coordinates of a point on the x-axis?

Its $y$ -coordinate is 0, for example $(5, 0)$ . A point on the $y$ -axis has $x$ -coordinate 0.

How do you check whether three points lie on one line?

Find the area of the triangle they make. If the area is 0, the points are collinear.

Where does the distance formula come from?

From Pythagoras' theorem: the horizontal and vertical gaps are the two shorter sides of a right-angled triangle.

Coordinate geometry

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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