Relations between Points and Lines

What this move does

Substitute the point $P_{0} (x_{0}, y_{0})$ into the normal form of a line and you get the signed distance $δ$ ; its absolute value is the distance. In terms of the general form $A x + B y + C = 0$ :

d = \frac{∣ A x _{0} + B y _{0} + C ∣}{A ^{2} + B ^{2}}, tan θ = \frac{A _{1} B _{2} - A _{2} B _{1}}{A _{1} A _{2} + B _{1} B _{2}} = \frac{k _{2} - k _{1}}{1 + k _{1} k _{2}} .

Here $θ$ is the angle through which $l_{1}$ must turn counterclockwise to reach $l_{2}$ .

Distance: substitute, take the absolute value, divide by $A^{2} + B^{2}$ .
Which side: points where $A x_{0} + B y_{0} + C$ has the same sign lie on the same side of the line. Comparing the signs for two points (or for a point and the origin) is all it takes.
Angle, parallel, perpendicular: $A_{1} B_{2} - A_{2} B_{1} = 0$ means parallel (or the same line); $A_{1} A_{2} + B_{1} B_{2} = 0$ means perpendicular; otherwise use $tan θ$ above.

Predict first

What is the distance from the point $(2, - 1)$ to the line $3 x - 4 y + 5 = 0$ ?

Choose an answer first; then you can see the result.

Work through one

Example: the angle bisectors of l₁: x − 2y + 2 = 0 and l₂: 2x + y − 6 = 0

Step 1 / 4

First, how do they sit?
$A_{1} A_{2} + B_{1} B_{2} = 1 \cdot 2 + (- 2) \cdot 1 = 0$ : the two lines are perpendicular, meeting at $(2, 2)$ .

$A_{1} A_{2} + B_{1} B_{2} = 0$
A point at equal distances
Set up the equation: equal distances to both lines
Which one bisects which angle? Look at the signs

“Same sign, same side” is a direct use of the sign of the signed distance from the previous stop: multiplying each expression by its own normalizing factor does not change whether the two signs agree.

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: $tan θ = \frac{k _{2} - k _{1}}{1 + k _{1} k _{2}}$ gives the angle

When both lines have slopes, the slope formula and the coefficient formula give the same $tan θ$ . Drag and watch the reading for $θ$ .

Practice

Exercise 1Basic

Find the distance from the point $(- 1, 2)$ to the line $4 x + 3 y - 12 = 0$ .

Exercise 2Advanced

Find the distance between the parallel lines $3 x + 4 y - 7 = 0$ and $6 x + 8 y + 1 = 0$ .

Exercise 3Advanced

Find the angle $θ$ from the line $l_{1} : y = 2 x + 1$ to the line $l_{2} : y = - 3 x + 4$ (in degrees).

Exercise 4Challenge

Find the reflection $P^{'}$ of the point $P (3, 5)$ in the line $x - y + 1 = 0$ . (In the form (a,b).)

What this move does

Work through one

Example: the angle bisectors of l₁: x − 2y + 2 = 0 and l₂: 2x + y − 6 = 0

When it fails

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Distance between parallel lines without matching the coefficients