SPI Mathematics · Free notes

Coordinate Geometry: SPI Mathematics

Coordinate Geometry has 30 questions in the SPI bank, and one of them, a centroid, is from the 2022 paper. Twenty-one are EASY, eight MODERATE and one HARD. Nearly every question is one formula with the numbers put in: distance, the section formula with its midpoint and centroid cases, area or slope. So the marks are lost on signs, not ideas: subtracting a negative coordinate, putting the ratio's m with the wrong end, or keeping one of two answers.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

SPI home All formulas on one page

Subtopic notes

Formula & revision sheet

9 formulas · 1 reference tables · 15 gotchas across all subtopics — the exam-eve cheat-sheet

Coordinates and the Distance Formula

Formulas (2)

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

Reference tables (1)

Axes, quadrants, abscissa and ordinate11 rows

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.

Watch out for (5)

y = k is parallel to the x-axis, not the y-axis
→ Axes, quadrants, abscissa and ordinate
Losing a minus sign
→ The distance formula
Stopping before the square root
→ The distance formula
Keeping only one root
→ Finding an unknown coordinate from a distance
Leaving the squared unknown in an equidistance question
→ Finding an unknown coordinate from a distance

Section Formula, Midpoint and Centroid

Formulas (3)

The section formula · Section formula $P = \left(\frac{mx_2 + nx_1}{m + n},\ \frac{my_2 + ny_1}{m + n}\right)$
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 $G = \left(\frac{x_1 + x_2 + x_3}{3},\ \frac{y_1 + y_2 + y_3}{3}\right)$

Watch out for (5)

Putting m with the wrong end
→ The section formula
Setting the wrong coordinate to zero
→ The section formula
Adding the wrong pair of corners
→ The midpoint and its uses
Dividing by 2 instead of 3
→ The centroid of a triangle
Forgetting to multiply back
→ The centroid of a triangle

Area, Slope and Straight Lines

Formulas (4)

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

Watch out for (5)

Forgetting the half
→ Area of a triangle from its corners
Subtracting in different orders
→ The slope of a line
Naming a triangle that is not there
→ Three points on one line
Sign slips with a negative point
→ The equation of a line
Checking only the slope
→ The equation of a line

Question weightage by concept

10 concepts · 30 questions — where the marks actually sit, so you know what to drill first

Coordinates and the Distance Formula11 questions · 37%

Concept	Qs	Share
Axes, quadrants, abscissa and ordinate	4	13%
The distance formula	4	13%
Finding an unknown coordinate from a distance	3	10%

Section Formula, Midpoint and Centroid12 questions · 40%

Concept	Qs	Share
The section formula	4	13%
The midpoint and its uses	4	13%
The centroid of a triangle	4	13%

Area, Slope and Straight Lines7 questions · 23%

Concept	Qs	Share
The equation of a line	3	10%
Area of a triangle from its corners	2	7%
The slope of a line	1	3%
Three points on one line	1	3%

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