SPI Mathematics · Coordinate Geometry

Area, Slope and Straight Lines

Three corners give a triangle's area by one formula, two points give a line's slope, three points lie on one line when the area they make is zero, and a slope with one point gives the line's equation.

All Coordinate Geometry notes

Why this matters

Seven questions in the bank, five EASY and two MODERATE: two areas, one slope, three points that turn out to lie on one line, and three equations of a line. The MODERATE ones hide a trap: a triangle made by lines (find its corners first) and a triangle that is not a triangle at all.

Concept 1 of 4: Area of a triangle from its corners

If a triangle has a side along an axis, half base times height is enough. For any three corners there is one formula that does the same job, and it never needs the height.

Definition

Area of the triangle with corners $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is $\frac{1}{2}\left|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\right|$ .
The bars mean: if the bracket comes out negative, drop the minus sign. An area is never negative.
A right triangle with legs along the axes, corners (0, 0), (a, 0) and (0, b), has area $\frac{1}{2}ab$ .
For a triangle made by three lines, first find its corners: each pair of lines meets at one corner. The line $y = x$ holds every point whose two coordinates are equal.

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|

Worked example

Find the area of the triangle with vertices (1, 1), (4, 1) and (4, 5).

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Coordinate Geometry · Area, Slope and Straight Lines

The area of the triangle formed by the lines x = 3, y = 4 and x = y is

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.

Drill 1 more on area of a triangle from its corners

Concept 2 of 4: The slope of a line

Slope is steepness: how far the line rises for each step to the right. Rise over run, taken between any two points on the line.

Definition

Slope through $(x_1, y_1)$ and $(x_2, y_2)$ : $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ .
A horizontal line has slope 0; a vertical line has no slope (the bottom would be 0).
Parallel lines have equal slopes.
The line $ax + by + c = 0$ has slope $-\dfrac{a}{b}$ .

Slope

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

Worked example

Find the slope of the line through (1, 2) and (5, 10).

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Coordinate Geometry · Area, Slope and Straight Lines

The slope of the line passing through the points A

(-4, -5)

and B

(-1, -2)

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.

Concept 3 of 4: Three points on one line

Three points on one straight line cannot make a triangle: the 'triangle' is flat, so its area is zero. Equally, the slope from the first point to the second matches the slope from the second to the third.

Definition

Points A, B and C are collinear (on one line) when any one of these holds:

the area of triangle ABC is 0;
slope AB = slope BC;
the two shorter distances add up to the longest: AB + BC = AC.

Before naming the type of a triangle from its corners, check that the points are not collinear.

Collinearity test

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

Worked example

Show that (1, 2), (3, 6) and (5, 10) lie on one line.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Coordinate Geometry · Area, Slope and Straight Lines

The points (2, 5), (4,

-

1) and (6,

-

7) are vertices of an/a

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.

Concept 4 of 4: The equation of a line

A line is fixed by its steepness and one point it passes through. Write the slope between that point and any point (x, y) on the line, and you have its equation.

Definition

Slope and y-intercept: $y = mx + c$ , where c is where the line cuts the y-axis.
Slope and a point $(x_1, y_1)$ : $y - y_1 = m(x - x_1)$ .
A line parallel to $ax + by + c = 0$ has the form $ax + by + k = 0$ ; put the given point in to find k.
Clear fractions and bring every term to one side to match the options' form.

Equation of a line

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

Worked example

Find the equation of the line through (1, 2) with slope 3.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Coordinate Geometry · Area, Slope and Straight Lines

Equation of the line passing through the point

(-2, -3)

and having slope

\frac{3}{5}

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

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.

Drill 2 more on the equation of a line

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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

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