1. The Cartesian coordinate plane

The plane is described by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). They meet at the origin, the point (0, 0). Any point is written as an ordered pair (x, y): the x-coordinate first, then the y-coordinate.

Read it aloud: "Point P has coordinates three comma two." — note that in English we say "three comma two", not "three two".

2. The equation of a straight line

The most common form of a line's equation is the gradient–intercept form:

y = mx + c

Read it: "y equals m x plus c."

• m is the gradient (also called the slope in American English) — how steep the line is.
• c is the y-intercept — where the line crosses the y-axis, i.e. the value of y when x = 0.

The gradient is the ratio of the rise to the run: how far the line goes up for every unit it goes right. Read it: "the slope equals the rise over the run."

The gradient from two points

If a line passes through two points (x₁, y₁) and (x₂, y₂), its gradient is

m = (y₂ − y₁) / (x₂ − x₁)

Read it: "m equals y two minus y one, over x two minus x one." — pronounced "y two", "x one", not "y subscript two".

3. Parallel and perpendicular lines

• Parallel lines have the same gradient: if m₁ = m₂, the lines never meet.
• Perpendicular lines have gradients that are negative reciprocals: m₁ · m₂ = −1. Read it: "the product of the two gradients is minus one."

4. The distance between two points

Given two points, the straight-line distance between them comes from Pythagoras' theorem:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Read it: "d equals the square root of x two minus x one, squared, plus y two minus y one, squared."

Think of it as a right-angled triangle: the horizontal gap is (x₂ − x₁), the vertical gap is (y₂ − y₁), and the distance is the hypotenuse. Read it: "the horizontal difference" and "the vertical difference".

5. The midpoint

The midpoint of the segment joining two points is the average of their coordinates:

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )

Read it: "M equals x one plus x two over two, y one plus y two over two." The midpoint bisects the segment — it divides it into two equal halves.

6. The equation of a circle

A circle is defined as the set of all points at a fixed distance — the radius — from a fixed point, the centre. For a circle centred at the origin with radius r:

x² + y² = r²

Read it: "x squared plus y squared equals r squared."

More generally, a circle with centre (a, b) and radius r is:

(x − a)² + (y − b)² = r²

Read it: "x minus a, squared, plus y minus b, squared, equals r squared." — This is the standard form of a circle's equation. You can always tell a circle from an equation: the x² and y² terms have equal coefficients and there is no xy term.

7. Tangents to a circle

A tangent is a line that touches the circle at exactly one point — the point of tangency. The tangent is always perpendicular to the radius drawn to that point. Read it: "The tangent is perpendicular to the radius at the point of contact."

8. Why coordinate geometry matters

Coordinate geometry is the bridge between algebra and geometry: it lets us represent shapes with equations and solve geometric problems with arithmetic. In English science writing, this subject appears in surprising places:

• Physics — the path of a projectile is a parabola in coordinates; velocity and acceleration are slopes of graphs.
• Computer graphics — every image on a screen is a grid of points with coordinates; lines and circles are drawn by their equations.
• Economics — supply and demand curves are graphed on coordinate planes; the equilibrium is their intersection point.

9. Quick check — test yourself

1. What are the gradient and y-intercept of y = −3x + 5? In what direction does the line slope?
2. A line passes through (1, 2) and (4, 8). Find its gradient. Is it steeper or less steep than y = 2x?
3. Give the centre and radius of (x + 1)² + (y − 4)² = 25.
4. Translate into English: "这条直线与 x 轴平行，因此它的斜率为零。"

Answers: 1. m = −3, c = 5; the line slopes downward (negative gradient). 2. m = (8−2)/(4−1) = 6/3 = 2; it has the same steepness as y = 2x. 3. Centre (−1, 4), radius 5. 4. "This line is parallel to the x-axis, so its gradient is zero."

10. Vocabulary recap