5.6 Parallel and Perpendicular Lines:

5.6 Parallel and Perpendicular Lines:

Opposite Reciprocals: Two numbers whose product is -1.

Parallel lines ( || ): Lines in the same plane with the same slope that never intersect.

Perpendicular lines ( ): Lines that intersect to form right angles and have opposite-inverse (opposite reciprocal) slopes.

GOAL:

Additional information we must know:Parallel Lines: Lines that have same slope and different y-intercept

y = x + 1 and y = x -2

Additional information we must know:Perpendicular Lines: Lines that have opposite and inverse slopes (opposite reciprocal slopes)

y = -2x + 1 and y = x -2

Checking:Looking at their slopes we see that: -2(1/2) = -1

WRITING AN EQUATION OF A PARALLEL LINE: Remember: Parallel lines we must have the same slope and different y-intercepts.

Ex:A line passes through (-3, -1) and is

parallel to the graph of y = 2x +3. What equation represents the line in slope-intercept form?

SOLUTION: Looking at the given info we have the following:

A line passes through (-3, -1) and is parallel to the graph of y = 2x +3. What equation

represents the line in slope-intercept form?

Point: (-3, -1) Slope: 2 Using point-slope {- = m(-)} form equation we get:

- = 2(--3) = 2(3)

= 2x+5 = 2)

YOU TRY IT:

What equation represents the line in slope-intercept form of a line passing through (-2, 1) and parallel to 2x+3y = 6

SOLUTION: Looking at the given info we have the following:

Point: (-2, 1) Slope: -

Using point-slope {- = m(-)} form equation we get:

- = - (--2)

-1 = - () = - x -

What equation represents the line in slope-intercept form of a line passing through (-2, 1) and parallel to 2x+3y = 6

y = - x + 2Finding the slope:

WRITING AN EQUATION OF PERPENDICULAR LINES: Remember: Perpendicular lines must have opposite and inverse slopes.

Ex:Write an equation of the line that

passes through (2, 4) and is perpendicular to y = x – 1.

SOLUTION: Looking at the given info we have the following:

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

Point: (2, 4) Slope: opposite inverse slope: -3

Using point-slope {- = m(-)} form equation we get:

- = -3(-2) - 4 = -3( - 2) = -3x+10

YOU TRY IT:

What equation represents the line in slope-intercept form of a line passing through (-2, 1) and perpendicular to 2x+3y = 6

SOLUTION: Looking at the given info we have the following:

Point: (-2, 1) Slope: - opposite inverse

Using point-slope {- = m(-)} form equation we get:

- = (--2)

-1 = () = x + 4

What equation represents the line in slope-intercept form of a line passing through (-2, 1) and perpendicular to 2x+3y = 6

y = - x + 2Finding the slope:

CLASSIFYING LINES: To classify a pair of lines we must look at their slope and y-intercepts thus we must write the equations in slope-intercept form (y=mx+b).

* Parallel lines must have the same slope and different y-intercepts.

*Perpendicular lines must have opposite and inverse slopes.

MUST ALWAYS REMEMBER:

CLASSIFYING LINES:

Ex:Decide if the given equations are

parallel, perpendicular or neither? Explain.

4y = -5x + 12 and 5x + 4y = -8

SOLUTION: Writing the equations in slope-intercept form to compare we have:1) 4y = -5x + 12

2) 5x + 4y = -8

y = -x + 3

Divide by 4

Slope: - , y-intercept: 3

Subtract 5x and divide by 4

y = -x - 2 Slope: - , y-intercept: - 2

Slopes are equal, and y-intercepts different, thus we have PARALLEL LINES.

YOU TRY IT:

Decide if the given equations are parallel, perpendicular or neither? Explain.

y = x + 7 and 4x - 3y = 9

YOU TRY IT (SOLUTION): Writing the equations in slope-intercept form to compare we have:

2) 4x - 3y = 9

Already in y=mx+b form

Slope: , y-intercept: 7

Subtract 4x and divide by - 3

y = x - 3 Slope: , y-intercept: - 3

Since the slopes are not equal or opposite reciprocals, the lines are neither.

y = x + 7

SOLVING REAL-WORLD PROBLEMS

In real-world situation we keep on using linear equations to provide important information that will aid us in making

important decisions.

EX: An architect uses

software to design the ceiling of a room. The architect needs to enter and equation that represents a new beam. The new beam will be perpendicular to the existing beam, which is represented the red line. The new beam will pass through the corner represented by the blue dot. What is an equation of the new beam?

SOLUTION:1) Find the slope of the red line:

Using A(3,6)and B(6,4) we find slope:

2) Find the opposite reciprocal slope of

3) Use the slope = and the point ( 12, 10)

In - = m(-) to get: - = (-)

- = - = -

VIDEOS:Parallel/Perpendicular

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/more-analytic-geometry/v/perpendicular-line-slope

https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/more-analytic-geometry/v/equations-of-parallel-and-perpendicular-lines

CLASSWORK:

Page 330-332

Problems: As many as needed to master the concept