Section 3.6 : Slopes of Parallel and Perpendicular Lines

Learning Targets: G.GPE.5, A.CED.2, F.IF.7a and F.LE.2

Important Terms and Definitions

Parallel Lines: Lines that are in the same plane and never intersect (Symbol: ∥)

Perpendicular Lines: Lines that intersect to form right angles (Symbol: ⊥)

Opposite Reciprocals: Opposite signs (+/−), and flipped fraction (23

, 32)

It’s All About Slope, Baby…

Quick Reminder

Horizontal lines have a slope of zero

Vertical lines have an undefined slope

New Info

Parallel lines have slopes that are the same

Perpendicular lines have slopes that are opposite reciprocals

(ex 1) Find the slope of the line parallel to and the slope of the line perpendicular to 𝑦 = 1

4𝑥 − 3.

(ex 2) Find the slope of the line parallel to and the slope of the line perpendicular to 2𝑥 + 3𝑦 = 15.

(ex 3) Find the slope of the line parallel to and the slope of the line perpendicular to 𝑦 = 4.

(ex 4) Are the lines 𝑦 = 2𝑥 and 𝑦 = −2𝑥 + 4 parallel, perpendicular, or neither?

(ex 5) Are the lines 𝑦 = 14

𝑥 + 5 and 𝑥 − 4𝑦 = 8 parallel, perpendicular, or neither?

(ex 6) Are the lines 3𝑦 = 6𝑥 + 4 and 𝑥 + 2𝑦 = 3 parallel, perpendicular, or neither?

(ex 7) Are the lines 𝑥 = 4 and 𝑦 = −1 parallel, perpendicular, or neither?

Writing Equations of Parallel and Perpendicular Lines

Step One: Find the new slope

Step Two: Use point-slope form with the new slope and the given point

Step Three: If necessary, solve for y to put the equation in slope-intercept form

Example: Write an equation of the line that is parallel to 𝑦 = 3𝑥 + 4 and passes through the point (−2, 5).

Step 1 : 𝑚 = 3 Step 2 : 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) Step 3 : 𝑦 − 5 = 3(𝑥 + 2)

𝑦 − 5 = 3(𝑥 + 2) 𝑦 − 5 = 3𝑥 + 6

𝑦 = 3𝑥 + 11

(ex 8) Write an equation of the line that is parallel to 3𝑥 + 4𝑦 = 20 and passes through (−8, −4).

(ex 9) Write an equation of the line that is parallel to 𝑥 = −4 and passes through (3, 5).

(ex 10) Write an equation of the line that is perpendicular to 𝑦 = − 12

𝑥 + 3 and passes through (−4, 5).

(ex 11) Write an equation of the line that is perpendicular to 𝑥 − 2𝑦 = 4 and passes through (1, −6).

(ex 12) A right triangle has two sides that are perpendicular to each other. Triangle XYZ has vertices 𝑋(4,3), 𝑌(2, −1), and 𝑍(0,1).Is XYZ a right triangle? Explain your reasoning.

Homework – Section 3.6 : Slopes of Parallel and Perpendicular Lines

Find the slope of the line parallel to and the slope of the line perpendicular to the given equation.

1. 𝑦 = 23

𝑥 + 5 4. 𝑥 + 9𝑦 = 1

2. 𝑦 = −4𝑥 − 12 5. 4𝑥 − 3𝑦 = 12

3. 2𝑥 + 𝑦 = 4 6. 𝑥 = −7

Are the lines parallel, perpendicular, or neither?

7. 𝑥 − 3𝑦 = −6 9. 6𝑥 = 5𝑦 + 1 3𝑥 − 𝑦 = 0 −12𝑥 + 10𝑦 = 1

8. −5𝑥 + 𝑦 = −6 10. 11𝑥 + 10𝑦 = 3 𝑥 + 5𝑦 = 5 11𝑦 = 10𝑥 + 5

Write an equation in slope-intercept form of the line that is parallel to the given line and passes through the given point.

11. 2𝑥 + 𝑦 = 8; (−1, −3) 12. 𝑥 − 2𝑦 = 6; (1, 4)

Write an equation in slope-intercept form of the line that is perpendicular to the given line and passes through the given point.

13. 3𝑥 − 2𝑦 = 10; (1, 1) 14. 𝑥 − 5𝑦 = 4; (−1, 1)

15. In a rectangle, opposite sides are parallel and adjacent sides are perpendicular. Figure ABCD has vertices 𝐴(−3, 3), 𝐵(−1, −2), 𝐶(4,0), and 𝐷(2,5). Show that ABCD is a rectangle.