Understanding Slope of Parallel and Perpendicular Lines in Geometry

Equation	Slope	$y -$ Intercept
$y = - 5 x + (- 3)$	$- 5$	$- 3$
$y = - 5 x + (- 1)$	$- 5$	$- 1$

Equation

Slope

y -

Intercept

y = - 5 x + (- 3)

- 5

- 3

y = - 5 x + (- 1)

- 5

- 1

Condition	Conclusion	Example
The lines of the system have the same slope and the same $y -$ intercept.	The lines are coincidental. This means that there are infinitely many points of intersection. Therefore, the system has infinitely many solutions.	${y = 3 x + 1 y = 3 x + 1$
The lines of the system have the same slope but different $y -$ intercept.	The lines are parallel. This means that there is not a point of intersection. Therefore, the system has no solution.	${y = 3 x + 1 y = 3 x + 5$
The lines of the system have different slopes.	The lines are neither parallel nor coincidental. This means that there is one point of intersection. Therefore, the system has one solution.	${y = 3 x + 1 y = - 2 x + 5$

Condition

Conclusion

Example

The lines of the system have the same slope and the same

y -

intercept.

The lines are coincidental. This means that there are infinitely many points of intersection. Therefore, the system has infinitely many solutions.

{y = 3 x + 1 y = 3 x + 1

The lines of the system have the same slope but different

y -

intercept.

The lines are parallel. This means that there is not a point of intersection. Therefore, the system has no solution.

{y = 3 x + 1 y = 3 x + 5

The lines of the system have different slopes.

The lines are neither parallel nor coincidental. This means that there is one point of intersection. Therefore, the system has one solution.

{y = 3 x + 1 y = - 2 x + 5

Since the theorem is a biconditional statement, the proof consists of two parts.

If two non-vertical lines are perpendicular, then the product of their slopes is $- 1 .$
If the product of the slopes of two non-vertical lines is $- 1,$ then the lines are perpendicular.

Part $1$

Let

ℓ_{1}

and

ℓ_{2}

be two perpendicular lines. Therefore, they intersect at one point. For simplicity, the lines will be translated so that the point of intersection is the origin.

Let

m_{1}

and

m_{2}

be the slopes of the lines

ℓ_{1}

and

ℓ_{2},

respectively. Next, consider the vertical line

x = 1 .

This line intersects both

ℓ_{1}

and

ℓ_{2} .

Since

ℓ_{1}

and

ℓ_{2}

are assumed to be perpendicular,

△ A O C

is a right triangle. Using the Distance Formula, the lengths of the sides of this triangle can be found.

Side	Points	$Distance Formula (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$	Length
$\overline{A O}$	$A (1, m_{1})$ $&$ $O (0, 0)$	$(0 - 1)^{2} + (0 - m_{1})^{2}$	$1 + m_{1}^{2}$
$\overline{C O}$	$C (1, m_{2})$ $&$ $O (0, 0)$	$(0 - 0)^{2} + (0 - m_{2})^{2}$	$1 + m_{2}^{2}$
$\overline{C A}$	$C (1, m_{2})$ $&$ $A (1, m_{1})$	$(1 - 1)^{2} + (m_{1} - m_{2})^{2}$	$m_{1} - m_{2}$