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Cambridge IGCSE Maths (0580) · Coordinate Geometry

Mini-Lesson

Coordinate Geometry

2 missions. Guess, learn, then check. Your place is saved.

Mission 1Rise 1 for every 12 along: which of three ramps is legal?
Mission 2Where can a mast stand that is equally far from two villages?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

Rise 1 for every 12 along: which of three ramps is legal?

Make a guess: Rise 1 for every 12 along: which of three ramps is legal?

Algebra · straight-line graphs

Gradient measures steepness

Gradient = change in $y$ y ÷ change in $x$ x ("rise over run").

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Uphill from left to right: positive. Downhill: negative. Horizontal: 0.
The bigger the gradient (ignoring its sign), the steeper the line.
"Rise 1 for every 12 along" is a gradient of $\frac{1}{12}$ 1/12.
Core: find the gradient by counting squares on a grid. Extended: also calculate it from two coordinates, as in the worked example below.

Algebra · straight-line graphs

The equation $y = m x + c$ y = mx + c

In $y = m x + c$ y = mx + c, $m$ m is the gradient and $c$ c is the $y$ y-intercept: the line crosses the $y$ y-axis at $(0, c)$ (0, c).
Rearrange to $y = \dots$ y = … first: $2 y = 6 x - 4$ 2y = 6x − 4 gives $y = 3 x - 2$ y = 3x − 2, so $m = 3$ m = 3 and $c = - 2$ c = −2.
To plot: make a table of values (three or more $x$ x values), plot the points, rule the line.
$y = k$ y = k is a horizontal line; $x = k$ x = k is a vertical line.
Extended also uses forms such as $3 x + 2 y = 5$ 3x + 2y = 5, which rearranges to $y = - 1.5 x + 2.5$ y = −1.5x + 2.5.

Quick check

?What is the gradient of the line

2 y = 12 - 3 x

Worked example Algebra · straight-line graphs

Equation of the line through two points

Question

Find the equation of the line through

(2, 7)

and

(5, 13)

1
Find the gradient: $m = \frac{13 - 7}{5 - 2} = \frac{6}{3} = 2$ m = (13 − 7)/(5 − 2) = 6/3 = 2
2
Substitute one point into $y = 2 x + c$ y = 2x + c: $7 = 2 \times 2 + c$ 7 = 2 × 2 + c
3
Solve for $c$ c: $c = 3$ c = 3
4
Check with the other point: $2 \times 5 + 3 = 13$ 2 × 5 + 3 = 13 ✓

Answer

y = 2 x + 3

⚠️ Watch out

subtract the coordinates in the same order on the top and the bottom, or the sign of the gradient flips.

Calculate

?A taxi fare is a fixed charge plus a set amount per mile, so fare against miles is a straight line. A 3-mile trip costs £8.40 and an 8-mile trip costs £15.40. Find the fixed charge. Give your answer as a decimal.

Algebra · straight-line graphs

Parallel and perpendicular lines

Parallel lines have equal gradients. The line parallel to $y = 4 x - 1$ y = 4x − 1 through $(1, - 3)$ (1, −3): $- 3 = 4 \times 1 + c$ −3 = 4 × 1 + c, so $y = 4 x - 7$ y = 4x − 7.
Extended: perpendicular lines have gradients that multiply to $- 1$ −1. Use the negative reciprocal: gradient 2 becomes $- \frac{1}{2}$ −1/2.
Extended: $2 y = 3 x + 1$ 2y = 3x + 1 has gradient $\frac{3}{2}$ 3/2, so a perpendicular line has gradient $- \frac{2}{3}$ −2/3.

Calculate

?(Extended) Line

L

has equation

y = 2 x + 5

. Line

M

is perpendicular to

L

and passes through

(4, 1)

. Find the

y

-intercept of line

M

Algebra · straight-line graphs

Midpoint and length of a line segment

Extended content.
The midpoint averages the coordinates:

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

The length comes from Pythagoras: $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$ √((x₂ − x₁)² + (y₂ − y₁)²).
For $A (1, 2)$ A(1, 2) and $B (7, 10)$ B(7, 10): the midpoint is $(4, 6)$ (4, 6) and the length is $6^{2} + 8^{2} = 10$ √(6² + 8²) = 10.

Calculate

?(Extended) Point

P

(- 3, 8)

and point

Q

(9, - 2)

. Find the length of

PQ

. Give your answer to 3 significant figures.

Mission check

?The rule: no steeper than a rise of 1 for every 12 along. Ramp P rises 0.6 m over 9 m, ramp Q is the line

y = 0.1 x

, ramp R rises 0.25 m over 3 m. Which ramps are legal?

Mission 2

Where can a mast stand that is equally far from two villages?

Make a guess: Where can a mast stand that is equally far from two villages?

Coordinate geometry · lines and bisectors

Drawing a line from its equation

Make a table of values: choose three $x$ x values, work out $y$ y, plot the points and rule a line through them. The third point is a check.
For $y = - 3 x + 2$ y = −3x + 2: $x = - 1, 0, 2$ x = −1, 0, 2 gives $y = 5, 2, - 4$ y = 5, 2, −4.
$x = k$ x = k is a vertical line; $y = k$ y = k is a horizontal line.
Extended: for $a x + b y = c$ ax + by = c, put $x = 0$ x = 0, then $y = 0$ y = 0, to find where the line crosses the axes. $3 x + 4 y = 12$ 3x + 4y = 12 crosses at $(0, 3)$ (0, 3) and $(4, 0)$ (4, 0).

Quick check

?Which point lies on the line

y = - 2 x + 5

Coordinate geometry · lines and bisectors

Lines in the form $a x + b y = c$ ax + by = c (Extended)

To find the gradient and $y$ y-intercept, make $y$ y the subject: $5 x + 4 y = 8$ 5x + 4y = 8 gives $4 y = - 5 x + 8$ 4y = −5x + 8, so $y = - 1.25 x + 2$ y = −1.25x + 2.
The gradient is $- 1.25$ −1.25 and the $y$ y-intercept is $(0, 2)$ (0, 2).
To go the other way, clear fractions, then collect $x$ x and $y$ y on one side: $y = \frac{2}{3} x - 4$ y = 2/3x − 4 gives $3 y = 2 x - 12$ 3y = 2x − 12, so $2 x - 3 y = 12$ 2x − 3y = 12.
Give equations fully simplified: $4 x + 6 y = 10$ 4x + 6y = 10 becomes $2 x + 3 y = 5$ 2x + 3y = 5.

Calculate

?Find the

y

-intercept of the line

6 x - 4 y = 9

. Give your answer as a decimal. (Extended)

Worked example Coordinate geometry · lines and bisectors

Equation in the form $a x + b y = c$ ax + by = c (Extended)

Question

Find the equation of the line through

(- 2, 5)

and

(4, 1)

. Give your answer in the form

a x + b y = c

, where

a

b

and

c

are integers.

1
Find the gradient: $m = \frac{1 - 5}{4 - ( - 2 )} = \frac{- 4}{6} = - \frac{2}{3}$ m = (1 − 5)/(4 − (−2)) = (−4)/6 = −2/3
2
Substitute $(4, 1)$ (4, 1) into $y = - \frac{2}{3} x + c$ y = −2/3x + c: $1 = - \frac{8}{3} + c$ 1 = −8/3 + c, so $c = \frac{11}{3}$ c = 11/3
3
Multiply every term by 3: $3 y = - 2 x + 11$ 3y = −2x + 11
4
Collect $x$ x and $y$ y on one side: $2 x + 3 y = 11$ 2x + 3y = 11
5
Check with $(- 2, 5)$ (−2, 5): $- 4 + 15 = 11$ −4 + 15 = 11 ✓

Answer

2 x + 3 y = 11

⚠️ Watch out

multiply every term by 3, including

c

. Writing

3 y = - 2 x + \frac{11}{3}

is a common slip.

Coordinate geometry · lines and bisectors

The perpendicular bisector (Extended)

The perpendicular bisector of $A B$ AB cuts $A B$ AB in half at right angles.
Every point on it is the same distance from $A$ A as from $B$ B.
To find its equation, first find the midpoint $M$ M of $A B$ AB.
Next, find the gradient of $A B$ AB, then the perpendicular gradient (the negative reciprocal).
Finally, substitute $M$ M into $y = m x + c$ y = mx + c with that gradient to find $c$ c.

Worked example Coordinate geometry · lines and bisectors

Finding a perpendicular bisector (Extended)

Question

A

(- 2, 1)

and

B

(4, 5)

. Find the equation of the perpendicular bisector of

A B

1
Midpoint: $M = (\frac{- 2 + 4}{2}, \frac{1 + 5}{2}) = (1, 3)$ M = ((−2 + 4)/2, (1 + 5)/2) = (1, 3)
2
Gradient of $A B$ AB: $\frac{5 - 1}{4 - ( - 2 )} = \frac{4}{6} = \frac{2}{3}$ (5 − 1)/(4 − (−2)) = 4/6 = 2/3, so the perpendicular gradient is $- \frac{3}{2}$ −3/2
3
Substitute $M (1, 3)$ M(1, 3) into $y = - \frac{3}{2} x + c$ y = −3/2x + c: $3 = - \frac{3}{2} + c$ 3 = −3/2 + c, so $c = 4.5$ c = 4.5
4
Clear the fraction: $2 y = - 3 x + 9$ 2y = −3x + 9, so $3 x + 2 y = 9$ 3x + 2y = 9

Answer

y = - 1.5 x + 4.5

, or

3 x + 2 y = 9

⚠️ Watch out

the line goes through the midpoint, not through

A

B

Calculate

?Find the equation of the perpendicular bisector of the line joining

(- 3, 9)

and

(7, 1)

. Write it in the form

y = m x + c

and enter the value of

c

as a decimal. (Extended)

Sort it

?Compare each line with

y = 2 x - 3

. Is it parallel, perpendicular or neither? (Extended)

Tap a statement, then tap the group it belongs in. Then press Check.

Mission check

?(Extended) Villages stand at

A (1, 2)

and

B (7, 6)

. Where can a mast go so that it is exactly as far from

A

as from

B

Recap

Big ideas

Gradient = change in $y$ y ÷ change in $x$ x; in $y = m x + c$ y = mx + c, $m$ m is the gradient and $(0, c)$ (0, c) the $y$ y-intercept.
To find a line's equation: gradient first, then substitute a point to find $c$ c.
Parallel lines share a gradient; perpendicular gradients multiply to $- 1$ −1 (Extended).
Midpoint: average the coordinates. Length: Pythagoras on the changes in $x$ x and $y$ y.
Draw a line from a table of values (three points); for $a x + b y = c$ ax + by = c, find where it crosses the axes (Extended).
To read the gradient and $y$ y-intercept of $a x + b y = c$ ax + by = c, make $y$ y the subject first (Extended).
Perpendicular bisector of $A B$ AB (Extended): through the midpoint, with the negative reciprocal of $A B$ AB's gradient. Every point on it is equally far from $A$ A and $B$ B.

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Mission 1Rise 1 for every 12 along: which of three ramps is legal?
Mission 2Where can a mast stand that is equally far from two villages?

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Specification points and sources for this lesson are listed at the bottom of the page.

Sources (2)

The specification points and sources behind this lesson.

1 Cambridge IGCSE Mathematics 0580 syllabus 2025–2027, C3.1–C3.6, E3.1–E3.7
2 Cambridge IGCSE Mathematics 0580 syllabus 2025–2027, Core C3.1–C3.7 and Extended E3.1–E3.7 (Coordinate geometry), including the notes and examples

Coordinate Geometry

Rise 1 for every 12 along: which of three ramps is legal?

Gradient measures steepness

The equation y=mx+cy = mx + c

Equation of the line through two points

Parallel and perpendicular lines

Midpoint and length of a line segment

Where can a mast stand that is equally far from two villages?

Drawing a line from its equation

Lines in the form ax+by=cax + by = c (Extended)

Equation in the form ax+by=cax + by = c (Extended)

The perpendicular bisector (Extended)

Finding a perpendicular bisector (Extended)

Big ideas

Mini-lesson complete!

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The equation $y = m x + c$ y = mx + c

Lines in the form $a x + b y = c$ ax + by = c (Extended)

Equation in the form $a x + b y = c$ ax + by = c (Extended)