Geometry · Unit 2
Write a line in slope-intercept form or point-slope form, including parallel lines, perpendicular lines, horizontal and vertical lines, and perpendicular bisectors.
Jump to section:
y − y₁ = m(x − x₁)
parallel: same m
perpendicular: negative reciprocal
Point-slope
y − y₁ = m(x − x₁).
Example: through (1, 3) with slope 2.
y − 3 = 2(x − 1)
Parallel and perpendicular
Parallel lines have the same slope. Example: a line parallel to y = 4x + 1 still has m = 4.
Perpendicular slopes are negative reciprocals. Example: a line perpendicular to y = 2x + 5 has m = −12.
Perpendicular bisector
Four steps, in order:
1. Midpoint of the segment.
2. Slope of the segment.
3. Negative reciprocal (the perpendicular slope).
4. Point-slope using the midpoint and that new slope.
Example: endpoints (1, 6) and (5, 2).
Midpoint (3, 4). Segment slope −1. Perpendicular slope 1.
y − 4 = 1(x − 3)
y = x + 1
Horizontal and vertical
Horizontal through (a, b): y = b. Vertical through (a, b): x = a.
A vertical line is perpendicular to a horizontal line.
horizontal
vertical
One point is the y-intercept. Use y = mx + b.
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 0 − 8
y₂ − y₁ = −8
x₂ − x₁ = 4 − 0
x₂ − x₁ = 4
m = −84
m = −2
The point (0, 8) is the y-intercept, so b = 8.
y = mx + b
y = −2x + 8
Answer: y = −2x + 8
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 10 − 4
y₂ − y₁ = 6
x₂ − x₁ = 2 − 0
x₂ − x₁ = 2
m = 62
m = 3
The point (0, 4) is the y-intercept, so b = 4.
y = mx + b
y = 3x + 4
Answer: y = 3x + 4
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 0 − (−5)
y₂ − y₁ = 5
x₂ − x₁ = 5 − 0
x₂ − x₁ = 5
m = 55
m = 1
The point (0, −5) is the y-intercept, so b = −5.
y = mx + b
y = x − 5
Answer: y = x − 5
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 1 − 7
y₂ − y₁ = −6
x₂ − x₁ = 2 − 0
x₂ − x₁ = 2
m = −62
m = −3
The point (0, 7) is the y-intercept, so b = 7.
y = mx + b
y = −3x + 7
Answer: y = −3x + 7
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = −6 − 2
y₂ − y₁ = −8
x₂ − x₁ = −4 − 0
x₂ − x₁ = −4
m = −8−4
m = 2
The point (0, 2) is the y-intercept, so b = 2.
y = mx + b
y = 2x + 2
Answer: y = 2x + 2
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 1 − 6
y₂ − y₁ = −5
x₂ − x₁ = 5 − 0
x₂ − x₁ = 5
m = −55
m = −1
The point (0, 6) is the y-intercept, so b = 6.
y = mx + b
y = −x + 6
Answer: y = −x + 6
The equation is in point-slope form. Write it as y = mx + b.
y − 1 = 4(x − 2)
y − 1 = 4x − 8
y = 4x − 7
Answer: y = 4x − 7
y − 5 = 2(x − 3)
y − 5 = 2x − 6
y = 2x − 1
Answer: y = 2x − 1
y + 4 = 3(x − 1)
y + 4 = 3x − 3
y = 3x − 7
Answer: y = 3x − 7
y − 6 = −2(x − 4)
y − 6 = −2x + 8
y = −2x + 14
Answer: y = −2x + 14
y + 3 = −1(x + 5)
y + 3 = −x − 5
y = −x − 8
Answer: y = −x − 8
y − 2 = 12(x − 6)
y − 2 = 12x − 3
y = 12x − 1
Answer: y = 12x − 1
Two points. Write y − y₁ = m(x − x₁).
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 11 − 3
y₂ − y₁ = 8
x₂ − x₁ = 5 − 1
x₂ − x₁ = 4
m = 84
m = 2
Use (1, 3).
y − y₁ = m(x − x₁)
y − 3 = 2(x − 1)
Answer: y − 3 = 2(x − 1)
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 10 − 1
y₂ − y₁ = 9
x₂ − x₁ = 5 − 2
x₂ − x₁ = 3
m = 93
m = 3
Use (2, 1).
y − y₁ = m(x − x₁)
y − 1 = 3(x − 2)
Answer: y − 1 = 3(x − 2)
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = −3 − 5
y₂ − y₁ = −8
x₂ − x₁ = 3 − (−1)
x₂ − x₁ = 4
m = −84
m = −2
Use (−1, 5).
y − y₁ = m(x − x₁)
y − 5 = −2(x − (−1))
y − 5 = −2(x + 1)
Answer: y − 5 = −2(x + 1)
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 6 − (−2)
y₂ − y₁ = 8
x₂ − x₁ = 6 − 4
x₂ − x₁ = 2
m = 82
m = 4
Use (4, −2).
y − y₁ = m(x − x₁)
y − (−2) = 4(x − 4)
y + 2 = 4(x − 4)
Answer: y + 2 = 4(x − 4)
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 0 − 9
y₂ − y₁ = −9
x₂ − x₁ = 4 − 1
x₂ − x₁ = 3
m = −93
m = −3
Use (1, 9).
y − y₁ = m(x − x₁)
y − 9 = −3(x − 1)
Answer: y − 9 = −3(x − 1)
m = y₂ − y₁x₂ − x₁
y₂ − y₁ = 11 − 1
y₂ − y₁ = 10
x₂ − x₁ = −4 − (−6)
x₂ − x₁ = 2
m = 102
m = 5
Use (−6, 1).
y − y₁ = m(x − x₁)
y − 1 = 5(x − (−6))
y − 1 = 5(x + 6)
Answer: y − 1 = 5(x + 6)
Keep the slope. Write the line through the point in slope-intercept form.
Parallel lines have the same slope.
m = 2
y − y₁ = m(x − x₁)
y − 5 = 2(x − 1)
y − 5 = 2x − 2
y = 2x + 3
Answer: y = 2x + 3
Parallel lines have the same slope.
m = 4
y − y₁ = m(x − x₁)
y − (−2) = 4(x − 5)
y + 2 = 4(x − 5)
y + 2 = 4x − 20
y = 4x − 22
Answer: y = 4x − 22
Parallel lines have the same slope.
m = −1
y − y₁ = m(x − x₁)
y − 6 = −1(x − (−3))
y − 6 = −1(x + 3)
y − 6 = −x − 3
y = −x + 3
Answer: y = −x + 3
Parallel lines have the same slope.
m = 25
y − y₁ = m(x − x₁)
y − 5 = 25(x − (−2))
y − 5 = 25(x + 2)
y − 5 = 25x + 45
y = 25x + 45 + 5
y = 25x + 45 + 255
y = 25x + 295
Answer: y = 25x + 295
Parallel lines have the same slope.
m = −3
y − y₁ = m(x − x₁)
y − 1 = −3(x − 4)
y − 1 = −3x + 12
y = −3x + 13
Answer: y = −3x + 13
Parallel lines have the same slope.
m = 12
y − y₁ = m(x − x₁)
y − (−4) = 12(x − 6)
y + 4 = 12(x − 6)
y + 4 = 12x − 3
y = 12x − 7
Answer: y = 12x − 7
Use the negative reciprocal. Write the line through the point in slope-intercept form.
The given slope is 2, so the perpendicular slope is −12.
y − y₁ = m(x − x₁)
y − (−1) = −12(x − 3)
y + 1 = −12(x − 3)
y + 1 = −12x + 32
y = −12x + 32 − 1
y = −12x + 32 − 22
y = −12x + 12
Answer: y = −12x + 12
The given slope is 13, so the perpendicular slope is −3.
y − y₁ = m(x − x₁)
y − 4 = −3(x − 6)
y − 4 = −3x + 18
y = −3x + 22
Answer: y = −3x + 22
The given slope is −4, so the perpendicular slope is 14.
y − y₁ = m(x − x₁)
y − 8 = 14(x − 1)
y − 8 = 14x − 14
y = 14x − 14 + 8
y = 14x − 14 + 324
y = 14x + 314
Answer: y = 14x + 314
The given slope is 5, so the perpendicular slope is −15.
y − y₁ = m(x − x₁)
y − 2 = −15(x − 10)
y − 2 = −15x + 2
y = −15x + 4
Answer: y = −15x + 4
The given slope is −12, so the perpendicular slope is 2.
y − y₁ = m(x − x₁)
y − (−6) = 2(x − 4)
y + 6 = 2(x − 4)
y + 6 = 2x − 8
y = 2x − 14
Answer: y = 2x − 14
The given slope is 34, so the perpendicular slope is −43.
y − y₁ = m(x − x₁)
y − 1 = −43(x − (−5))
y − 1 = −43(x + 5)
y − 1 = −43x − 203
y = −43x − 203 + 1
y = −43x − 203 + 33
y = −43x − 173
Answer: y = −43x − 173
The given line is horizontal or vertical. Write the perpendicular line through the point.
x = 5 is vertical, so the perpendicular line is horizontal.
y = b
The line passes through (−4, −3), so b = −3.
y = −3
Answer: y = −3
x = −1 is vertical, so the perpendicular line is horizontal.
y = b
The line passes through (2, 7), so b = 7.
y = 7
Answer: y = 7
y = 4 is horizontal, so the perpendicular line is vertical.
x = a
The line passes through (5, −2), so a = 5.
x = 5
Answer: x = 5
y = −8 is horizontal, so the perpendicular line is vertical.
x = a
The line passes through (−6, 3), so a = −6.
x = −6
Answer: x = −6
x = 9 is vertical, so the perpendicular line is horizontal.
y = b
The line passes through (0, −5), so b = −5.
y = −5
Answer: y = −5
y = 6 is horizontal, so the perpendicular line is vertical.
x = a
The line passes through (8, 1), so a = 8.
x = 8
Answer: x = 8
Find the midpoint, then the perpendicular slope, then the equation.
midpoint = (x₁ + x₂2, y₁ + y₂2)
midpoint = (3, 4)
m = y₂ − y₁x₂ − x₁
m = 2 − 65 − 1 = −44 = −1
Perpendicular slope is 1.
y − 4 = 1(x − 3)
y = x + 1
Answer: y = x + 1
midpoint = (4, 4)
m = 6 − 28 − 0 = 48 = 12
Perpendicular slope is −2.
y − 4 = −2(x − 4)
y − 4 = −2x + 8
y = −2x + 12
Answer: y = −2x + 12
midpoint = (−1, 1)
m = −3 − 51 − (−3) = −84 = −2
Perpendicular slope is 12.
y − 1 = 12(x − (−1))
y − 1 = 12(x + 1)
Answer: y − 1 = 12(x + 1)
midpoint = (2, 2)
The segment is vertical, so the perpendicular bisector is horizontal.
y = 2
Answer: y = 2
midpoint = (x₁ + x₂2, y₁ + y₂2)
midpoint = (1 + 72, 1 + 52)
midpoint = (82, 62)
midpoint = (4, 3)
m = y₂ − y₁x₂ − x₁
m = 5 − 17 − 1
m = 46
m = 23
Perpendicular slope is −32.
y − y₁ = m(x − x₁)
y − 3 = −32(x − 4)
y − 3 = −32x + 6
y = −32x + 6 + 3
y = −32x + 9
Answer: y = −32x + 9
midpoint = (−1, −2)
m = −5 − 14 − (−6) = −610 = −35
Perpendicular slope is 53.
y − (−2) = 53(x − (−1))
y + 2 = 53(x + 1)
Answer: y + 2 = 53(x + 1)
Need help writing the equation of a line?
Book a Session with Amy