SME
Grade 10 Math · Problem type
A perpendicular line through a point
Use this for “Write the equation of the line perpendicular to y = 2x + 1 through (4, 3).”
Chart
Given: y = 2x + 1
Perpendicular: y = (−1/2)x + 5
They cross at (8/5, 4.2).
2 × 8/5 + 1 = 4.2
−1/2 × 8/5 + 5 = 4.2
The square marks a right angle: 2 × (−1/2) = −1
(4, 3) is on Perpendicular: −1/2 × 4 + 5 = 3
This is an example. Type your own values to start a new problem.
y
=
m₁
x
+
b₁
y
=
−1
2
m₂
x
+
b
x₀
Point, x
example
y₀
Point, y
example
Formulas
m₂
=
−
1
m₁
−
1
2
=
−
1
2
b
=
y₀
−
m₂
×
x₀
5
=
3
−
(−
1
2
)
×
4
Assumptions
•
Perpendicular slopes multiply to −1, so the new slope is −1 ÷ m.
•
Put the point into y = mx + b to find the new intercept.
•
A level line’s perpendicular is vertical, x = x₀: it has no slope.
Step-by-step
WE KNOW
m₁ = 2, b₁ = 1, x₀ = 4, y₀ = 3
FIND
perpendicular slope (m₂), new line’s intercept (b)
Step 1 · Find perpendicular slope (m₂)
Use m₂ = −1 ÷ m₁
Perpendicular slopes multiply to −1.
m₂
=
−
1
2
m₂
=
−
1
2
Step 2 · Find new line’s intercept (b)
Use b = y₀ − m₂ × x₀
The
point
is
on
the
new
line,
so
y₀
=
m₂
×
x₀
+
b
.
b
=
3
−
(−
1
2
)
×
4
b
=
3
−
(−2)
b
=
5
Check
Put the numbers back in. Both sides should match.
−
1
2
=
−
1
2
✓
5
=
3
−
(−
1
2
)
×
4
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Related lessons
Parallel and perpendicular lines: transversals and proofs
Main lesson
Alternate interior angles
Use this for “ℓ ∥ m and m∠1 = 118°. Find m∠3 and m∠6.”
Same-side interior angles
Use this for “ℓ ∥ m and m∠4 = 73°. Find m∠6.”
Are the lines parallel?
Use this for “m∠1 = 81° and m∠5 = 79°. Are lines ℓ and m parallel?”
Angle expressions
Use this for “ℓ ∥ m, corresponding angles are (4x + 12)° and (2x + 50)°. Find x and the angle.”
A parallel line through a point
Use this for “Write the equation of the line parallel to y = 3x − 4 through (2, 7).”
Why a triangle’s angles add to 180°
Use this for “Prove that the angles of a triangle add to 180°.”