Lesson 4-7

Example 1    Parallel Line Through a Given Point
Write the slope-intercept form of an equation for the line that passes through (-4, 1) and is parallel to the graph of y = x + 3.
The line parallel to y = x + 3 has the same slope, 1. Replace m with 1 and (x, y) with (-4, 1) in the point-slope form.
	y - y = m(x - x)	Point-slope form
	 y – 1 = 1(x – (-4))	Replace m with 1, y1 with 1, and x1 with –4.
	 y – 1 = 1(x + 4)	Simplify.
	 y – 1 = x + 4	Distributive Property
	 y – 1 + 1 = x + 4 + 1	Add 1 to each side.
		y = x + 5	Write equation in slope-intercept form.

Example 2    Determine Whether Lines are Perpendicular

GEOMETRY  If all four angles in a quadrilateral are right angles, the figure is a rectangle. Determine if A, B, C, and D are right angles and if quadrilateral ABCD is a rectangle.	

If , then D is a right angle.
Find the slopes of  and .
slope of  =  = 
slope of  =  = -
The lines and are perpendicular because · - = -1. Therefore, D is a right angle.

Find the slopes ofand .
slope of =  = 
slope of  =  = -
The lines and are perpendicular because · - = -1. Therefore, C is a right angle.

Find the slopes of and .
slope of =  = 
slope of =  = -
The lines  and  are perpendicular because · -  = -1. Therefore, B is a right angle.

Find the slopes of and .
slope of  =  = 
slope of  =  = -
The lines  and  are perpendicular because · - = -1. Therefore, A is a right angle.
So, all four angles are right angles and the figure is a rectangle.

Example 3    Perpendicular Line Through a Given Point
Write an equation in slope-intercept form for a line that passes through (-3, 4) and is perpendicular to the graph of 9x – 6y = 3.
Step 1: 	Find the slope of the given line.
	9x – 6y	= 3	Original equation
	  9x – 6y – 9x 	= 3 – 9x	Subtract 9x from each side.
	-6y 	= -9x + 3	Simplify.
	  	=	Divide each side by –6.
	   y 	= x - 	Simplify.

Step 2:	The slope of the given line is . So, the slope of the line perpendicular to this line is the opposite reciprocal of , or -.

Step 3:	Use the point-slope form to find the equation.
	y - y	= m(x - x)	Point-slope form
	 y - 4	= - (x – (-3))	(x1, y1) = (-3, 4) and m = -
	 y - 4	= - (x + 3)	Simplify.
	 y - 4	= -x + (-2)	Distributive Property
y - 4 + 4 	= -x + (-2) + 4	Add 4 to each side.
	   y	= -x + 2	Simplify.

Example 4    Perpendicular Line Through a Given Point
Write an equation in slope-intercept form for a line perpendicular to the graph of y = x –5 and passes through the y-intercept of that line.
Step 1:	Find the slope of the perpendicular line. The slope of the given line is , therefore, a perpendicular line has slope - because  · - = -1.

Step 2:	Find the y-intercept of the given line. 
y = x –5	Original equation
y = (0) – 5	Replace x with 0.
y = -5	Simplify.
The y-intercept is at (0, -5).

Step 3:	Substitute the slope and the given point into the slope-intercept form of an equation since you have both the slope and y-intercept.
y = mx + b	Slope-intercept form		
y = -x +(-5)	m = -and b = -5
y = -x – 5 	Simplify.