Coordinate Geometry — SPI Mathematics Formulas and Traps

Coordinates and the Distance Formula

The distance formula

Distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad OP = \sqrt{x^2 + y^2}

Finding an unknown coordinate from a distance

Equal distances

PA = PB \Rightarrow PA^2 = PB^2

Axes, quadrants, abscissa and ordinate

Term	Meaning	Example
Origin	Where the axes meet	(0, 0)
Abscissa	The x-coordinate	Abscissa of (−3, 5) is −3
Ordinate	The y-coordinate	Ordinate of (−3, 5) is 5
Quadrant I, II, III, IV	Signs (+, +), (−, +), (−, −), (+, −)	(−3, 5) is in quadrant II
Point on the x-axis	y-coordinate is 0	(4, 0)
Point on the y-axis	x-coordinate is 0	(0, −2)
Equation of the x-axis	y = 0	Every point (x, 0)
Equation of the y-axis	x = 0	Every point (0, y)
Line y = k	Parallel to the x-axis, k units from it	y = 3 passes through (1, 3) and (7, 3)
Line x = k	Parallel to the y-axis, k units from it	x = −2 passes through (−2, 0) and (−2, 6)
Distance from the x-axis	The size of y	(6, −4) is 4 units from the x-axis Distance from the x-axis is the size of y, and from the y-axis the size of x: the other coordinate.

Quadrants are counted anticlockwise from the top right.

Common traps

y = k is parallel to the x-axis, not the y-axis

The line y = 3 keeps y fixed while x runs, so it is horizontal: parallel to the x-axis. It is tempting to pair 'y' with the y-axis, which is the wrong way round.

Losing a minus sign

Subtracting a negative coordinate adds:

4 - (-2) = 6

, not 2. A wrong difference still gives a neat-looking square root, so check each difference before squaring.

Stopping before the square root

36 + 64 = 100

is the square of the distance. The distance is

\sqrt{100} = 10

; the 100 is often printed as a wrong option.

Keeping only one root

(y - 2)^2 = 16

gives

y - 2 = \pm 4

, so y is 6 or −2. Options often print both values together; a single value is incomplete.

Leaving the squared unknown in an equidistance question

When you set

PA^2 = PB^2

, the

x^2

(or

y^2

) appears on both sides and cancels, leaving a simple equation. If it does not cancel, a square has been expanded wrongly.

Section Formula, Midpoint and Centroid

Learn this subtopic in the notes

The section formula

Section formula

P = \left(\frac{mx_2 + nx_1}{m + n},\ \frac{my_2 + ny_1}{m + n}\right)

$m : n$ the ratio AP : PB
$(x_1, y_1)$ the first end, A
$(x_2, y_2)$ the second end, B

The midpoint and its uses

Midpoint formula

M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right) \qquad D = A + C - B

The centroid of a triangle

Centroid

G = \left(\frac{x_1 + x_2 + x_3}{3},\ \frac{y_1 + y_2 + y_3}{3}\right)

Common traps

Putting m with the wrong end

In

\frac{mx_2 + nx_1}{m + n}

, m multiplies the SECOND point. Swapping them gives the point that divides the join in the ratio n : m, and that point is usually one of the wrong options.

Setting the wrong coordinate to zero

A point on the x-axis has y = 0, so for 'the x-axis divides the join' solve the y-equation. Using the x-equation answers a different question.

Adding the wrong pair of corners

In parallelogram ABCD the diagonals are AC and BD, so D = A + C − B. Writing D = A + B − C pairs corners that are next to each other, and gives a point that makes a different shape.

Dividing by 2 instead of 3

The centroid averages THREE corners, so divide the sum by 3. Dividing by 2 is the midpoint habit, and the point it gives is not the centroid.

Forgetting to multiply back

To find a missing vertex, multiply the centroid by 3 first: the three x values must add to 3 times the centroid's x. Subtracting the known vertices straight from the centroid gives the wrong point.

Area, Slope and Straight Lines

Learn this subtopic in the notes

Area of a triangle from its corners

Area of a triangle

\text{Area} = \frac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|

The slope of a line

Slope

m = \frac{y_2 - y_1}{x_2 - x_1} \qquad ax + by + c = 0 \Rightarrow m = -\frac{a}{b}

Three points on one line

Collinearity test

x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0

The equation of a line

Equation of a line

y = mx + c \qquad y - y_1 = m(x - x_1)

Common traps

Forgetting the half

The bracket in the area formula gives TWICE the area. Without the

\frac{1}{2}

the answer is double, and the double is often printed as an option.

Subtracting in different orders

Take both differences from the same point:

\frac{y_2 - y_1}{x_2 - x_1}

. Mixing

y_2 - y_1

with

x_1 - x_2

flips the sign of the slope.

Naming a triangle that is not there

Three points with equal slopes between them lie on one line, so they make no triangle at all. When 'none of these' is an option beside isosceles, equilateral and right-angled, test for collinearity first.

Sign slips with a negative point

Through

(-4, -1)

the form is

y + 1 = m(x + 4)

: subtracting a negative coordinate turns into a plus. Writing

y - 1

or

x - 4

puts the line through a different point.

Checking only the slope

Several options can share the right slope. After finding the equation, put the given point into it: only the right line gives 0.

Coordinate Geometry formulas

Coordinates and the Distance Formula

The distance formula

Finding an unknown coordinate from a distance

Axes, quadrants, abscissa and ordinate

Common traps

Section Formula, Midpoint and Centroid

The section formula

The midpoint and its uses

The centroid of a triangle

Common traps

Area, Slope and Straight Lines

Area of a triangle from its corners

The slope of a line

Three points on one line

The equation of a line

Common traps

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