Lesson 10 of 10 · 14 min
Chapter review
Watch a class
The whole chapter on YouTube
Must-know facts
16 facts
- 01Distance: √((x₂ − x₁)² + (y₂ − y₁)²); from (6, −4) to (3, 0) it is 5.
- 02Section point (m : n internal): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
- 03Triangle area ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; zero means collinear.
- 04Inclination θ: 0° ≤ θ ≤ 180°; slope m = tan θ, not defined at θ = 90°.
- 05Slope through two points: m = (y₂ − y₁)/(x₂ − x₁).
- 06Parallel: m₁ = m₂. Perpendicular: m₁m₂ = −1.
- 07Acute angle between lines: tan θ = |(m₂ − m₁)/(1 + m₁m₂)|.
- 08Horizontal line y = a; vertical line x = b.
- 09Point-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
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.