Exercise 6 Page 227

Two lines are parallel if their slopes are identical. To tell if two lines are perpendicular, we have to check if their slopes are negative reciprocals. Let's tackle these questions one at a time.

Are They Parallel?

For this exercise, we have been given two points on each line, so we have enough information to calculate their slopes using the Slope Formula. m=y_2- y_1/x_2- x_1 Note that when choosing points to substitute for ( x_1, y_1) and ( x_2, y_2), it does not matter which points on the line you choose, since the result will be the same. Let's start with line a, which passes through ( 0, 4) and ( 4, 3).

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 0,4) & ( 4,3)

m=3- 4/4- 0

SubTerms

Subtract terms

m=-1/4

MoveNegNumToFrac

Put minus sign in front of fraction

m=-1/4

We will use a similar method to identify the slopes of lines b and c.

Line	Points	y_2-y_1/x_2-x_1	Slope
a	( 0, 4) & ( 4, 3)	3- 4/4- 0	-1/4
b	( 0, 1) & ( 4, 0)	0- 1/4- 0	-1/4
c	( 2, 0) & ( 4, 4)	4- 0/4- 2	2

Now that we have identified the slope of each line, we can see that a and b have the same slope. Therefore, line a and line b are parallel.

Are They Perpendicular?

For lines with different slopes, we can conclude that they are not parallel. To determine whether or not they are perpendicular, we will calculate the product of their slopes. Any two slopes whose product equals -1 are negative reciprocals, and are therefore perpendicular. m_1* m_2 ? =-1 Since only line c has a different slope, let's calculate the product of its slope and the slope of a and b.

m_1* m_2? =- 1

SubstituteII

m_1= -1/4, m_2= 2

( -1/4)( 2)? =- 1

MoveRightFacToNumOne

1/b* a = a/b

-2/4? =- 1

ReduceFrac

a/b=.a /2./.b /2.

-1/2≠- 1 *

Line c is neither parallel nor perpendicular to lines a and b. Therefore, none of the lines are perpendicular.