Understanding Parallel and Perpendicular Lines: Equations Explored

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

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}

Consider the following lines.

y_{1} = 4 x + 3 y_{3} = 3 + 5 x y_{5} = 4 + 4 x y_{2} = x + 2 y_{4} = 1 - 4 x y_{6} = - x + 5

Which of the lines are parallel?

We begin by recalling that parallel lines have the same slope. In order to determine which lines are parallel we will find the slope of each line. We can find the slope of a line by inspecting the coefficient of the x-term. Let's do it!

Line	Slope
y_1= 4x+3	4
y_2= x+2	1
y_3=3+ 5x	5
y_4=1 -4x	- 4
y_5=4+ 4x	4
y_6= - x+5	-1

In the table above we can see that the lines y_1 and y_5 both have a slope of 4. All other lines have different slopes. Therefore, y_1 and y_5 are the only parallel lines.

Consider the following diagram.

Let

L_{1}

be the line that passes through

A

and

B,

and let

L_{2}

be the line that passes through

C

and

D .

Are

L_{1}

and

L_{2}

parallel?

Let's begin by drawing lines L_1 and L_2.

At first glance the lines appear to be parallel. Let's be sure to find the slope of each line. Recall the Slope Formula. m=y_2-y_1/x_2-x_1 We first need to identify the coordinates of the given points. A=(- 5,- 1) B=(4,7) [0.5em] C=(- 4,- 5) D=(7,5) Next, substitute the coordinates of A and B to find the slope m_1 of L_1.

We find the slope m_2 of L_2 in a similar way.

The slopes of L_1 and L_2 are different. Therefore, the lines are not parallel.

Determine whether the following statement is always, sometimes, or never true.

Two lines with negative slopes are perpendicular.

Let's begin by recalling the relation between the slopes of two perpendicular lines.

Slopes of Perpendicular Lines Theorem |- In a coordinate plane, two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals.

The above theorem states that if non-vertical lines are perpendicular, the product of their slopes is always -1. The given statement tells us that both lines have negative slopes, so their product will be positive. Since the product is positive it cannot be -1, so the statement is never true.

Consider the following diagram.

Let

L_{1}

be the line that passes through

A

and

B,

and let

L_{2}

be the line that passes through

C

and

D .

Are

L_{1}

and

L_{2}

perpendicular?

Let's begin by drawing lines L_1 and L_2.

At first glance the lines appear to be perpendicular. Let's be sure by finding the slope of each line. Recall the Slope Formula. m=y_2-y_1/x_2-x_1 We first need to identify the coordinates of the given points. &A=(- 2,- 1) &&B=(2,1) [0.5em] &C=(1,- 2) &&D=(-1,2) Next, substitute the coordinates of A and B to find the slope m_1 of L_1.

We find the slope m_2 of L_2 in a similar way.

The Slopes of Perpendicular Lines Theorem tells us that two lines are parallel if and only if their slopes are negative reciprocals. Let's check! m_1=1/2 m_2=-2 [0.5em] m_1* m_2=- 1 ✓ We can see that 12 and -2 are negative reciprocals. Therefore, the lines are perpendicular.

Parallel and Perpendicular Lines

Catch-Up and Review

Observing Parallel Lines

Parallel Lines

Slopes of Parallel Lines Theorem

Proof

Part $1$

Part $2$

Identifying Parallel Lines

Finding a Parallel Line Through a Point

Hint

Solution

Bike Path

Hint

Solution

Starry Scene

Hint

Solution

Observing Perpendicular Lines

Perpendicular Lines

Slopes of Perpendicular Lines Theorem

Proof

Part $1$

Part $2$

Identifying Perpendicular Lines

Finding a Perpendicular Line Through a Point

Hint

Solution

Water Supply Network

Hint

Solution

Space Planning

Hint

Solution

Lines in the Three-Dimensional Space

Parallel and Perpendicular Lines

Recommended exercises

	14 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Parallel and Perpendicular Lines

Catch-Up and Review

Proof

Part 1

Part 2

Hint

Solution

Hint

Solution

Hint

Solution

Proof

Part 1

Part 2

Hint

Solution

Hint

Solution

Hint

Solution

Parallel and Perpendicular Lines

Recommended exercises

Part $1$

Part $2$

Part $1$

Part $2$