Lesson 10 of 10 · 14 min

Chapter review

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Must-know facts

16 facts

1Distance: √((x₂ − x₁)² + (y₂ − y₁)²); from (6, −4) to (3, 0) it is 5.
2Section point (m : n internal): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
3Triangle area ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; zero means collinear.
4Inclination θ: 0° ≤ θ ≤ 180°; slope m = tan θ, not defined at θ = 90°.
5Slope through two points: m = (y₂ − y₁)/(x₂ − x₁).
6Parallel: m₁ = m₂. Perpendicular: m₁m₂ = −1.
7Acute angle between lines: tan θ = |(m₂ − m₁)/(1 + m₁m₂)|.
8Horizontal line y = a; vertical line x = b.
9Point-slope form: y − y₀ = m(x − x₀).
10Two-point form: y − y₁ = ((y₂ − y₁)/(x₂ − x₁))(x − x₁).
11Slope-intercept form: y = mx + c; with x-intercept d: y = m(x − d).
12Intercept form: x/a + y/b = 1.
13General equation: Ax + By + C = 0, A and B not both zero.
14Point to line: d = |Ax₁ + By₁ + C|/√(A² + B²).
15Parallel lines: d = |C₁ − C₂|/√(A² + B²) = |c₁ − c₂|/√(1 + m²).
16(3, −5) is 3/5 from 3x − 4y − 26 = 0; 3x − 4y + 7 = 0 and 3x − 4y + 5 = 0 are 2/5 apart.

Common traps

Where marks are lost

Giving a vertical line slope 0.

A vertical line has θ = 90° and no slope; slope 0 belongs to a horizontal line.

Mixing the order of points: (y₂ − y₁)/(x₁ − x₂).

Take the same point first on top and bottom; mixing flips the sign of the slope.

Using m₁m₂ = −1 for a horizontal and a vertical line.

One slope is undefined, so check such pairs directly: y = a and x = b are always perpendicular.

Keeping only one answer when a line makes a given angle with another.

The modulus in tan θ = |(m₂ − m₁)/(1 + m₁m₂)| gives two slopes, as in m = 3 or −1/3.

Using the parallel-lines formula before matching the coefficients.

Write both lines with the same A and B first: 6x + 4y − 14 = 0 must become 3x + 2y − 7 = 0 before comparing with 3x + 2y + 6 = 0.

Leaving out √(A² + B²) or the modulus in the distance formula.

d = |Ax₁ + By₁ + C|/√(A² + B²) is a length, so it is never negative.

Reading c in y = mx + c as the x-intercept.

c is where the line meets the y-axis; the x-intercept of y = mx + c is −c/m.

Taking a distance along a slanted line as the perpendicular distance.

Along a given direction, find the meeting point and use the distance formula; the perpendicular distance is the shortest one.

Formulas

11 to know

Distance between two points

√((x₂ − x₁)² + (y₂ − y₁)²)

Section formula (internal, m : n)

((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))

m = n gives the mid-point.

Area of a triangle

½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Zero for collinear points.

Slope

m = tan θ = (y₂ − y₁)/(x₂ − x₁)

θ ≠ 90°, x₁ ≠ x₂.

Angle between lines

tan θ = |(m₂ − m₁)/(1 + m₁m₂)|

Acute angle; 1 + m₁m₂ ≠ 0.

Point-slope form

y − y₀ = m(x − x₀)

Two-point form

y − y₁ = ((y₂ − y₁)/(x₂ − x₁))(x − x₁)

Slope-intercept form

y = mx + c or y = m(x − d)

c is the y-intercept, d the x-intercept.

Intercept form

x/a + y/b = 1

Intercepts a on the x-axis and b on the y-axis.

Point to line

d = |Ax₁ + By₁ + C|/√(A² + B²)

Between parallel lines

d = |C₁ − C₂|/√(A² + B²)

Same A and B in both equations.

Key terms

11 terms

Inclination: The angle θ, from 0° to 180°, that a line makes anticlockwise with the positive x-axis.
Slope: tan θ for a non-vertical line; the rise per unit run.
Collinear points: Points that lie on one straight line.
y-intercept: The value c where a line meets the y-axis at (0, c).
x-intercept: The value where a line meets the x-axis.
Point-slope form: y − y₀ = m(x − x₀), the line through (x₀, y₀) with slope m.
Intercept form: x/a + y/b = 1, the line with intercepts a and b.
General equation of a line: Ax + By + C = 0 with A and B not both zero.
Concurrent lines: Three or more lines passing through one common point.
Perpendicular distance: The length of the perpendicular from a point to a line: the shortest distance between them.
Image of a point: The reflection of a point in a line, which acts as the perpendicular bisector of the point and its image.