Perpendicular Lines

Perpendicular Lines

How to solve perpendicular lines problems (linear equations): definition, formula, examples, and their solutions.

Definition, Formula

Perpendicular lines are lines that form a right angle. (m1)*(m2) = -1. m1, m2: Perpendicular lines' slopes

Perpendicular lines are lines
that form a right angle (= 90º).

The red symbol is the right angle symbol.
It shows that
the intercepted lines are perpendicular.

If two perpendicular lines' slopes are m1 and m2,
then m1m2 = -1.

Example 1

If y = qx - 2 is perpendicular to the given line, find the value of q.

See the given line.

The change of x is 2.
And the change of y is 1.

So the slope of the line is
m = 1/2.

Slope of a line

And it says
the given line [slope: 1/2]
and y = qx - 2 [slope: q]
are perpendicular lines.

So (1/2)⋅q= -1.

Multiply 2 on both sides.

Then q = -2.

Let's see the graph of the parallel lines.

The increasing line is the given line.

And the decreasing line is y = qx - 2,
which is y = -2x - 2.

Its slope is -2.
And its y-intercept is -2.
So you can easily draw y = -2x - 2.

Slope-intercept form - Example 2.

These two lines are perpendicular.
So you can use the right angle symbol (red).

Example 2

Find the linear equation that is perpendicular to y = 3x + 4 and that passes through (3, 1).

The slope of y = 3x + 4 is 3.

So set m1 = 3.

It says
the linear equation is perpendicular to y = 3x + 4.
[slope: 3]

Set m2 as the slope of the linear equation.
[slope: m2]

Then 3⋅m2 = -1.

Multiply 3 on both sides.
Then m2 = -1/3.

The slope of the line is -1/3.

And it says
the line passes through (3, 1).

Then the linear equation in point-slope form
is y = (-1/3)(x - 3) + 1.

Change the linear equation
in slope-intercept form.

Then y = (-1/3)x + 2.

Let's see the graph of the perpendicular lines.

The increasing line is the given line:
y = 3x + 4.

And the decreasing line is the answer:
y = (-1/3)x + 2.

It is perpendicular to the given line.
And it passes through (3, 1).

These two lines are perpendicular.
So you can use the right angle symbol (red).