Answer:
Part A) [tex]y=\frac{3}{4}x-\frac{1}{4}[/tex]
Part B) [tex]y=\frac{2}{7}x-\frac{5}{7}[/tex]
Part C) [tex]y=\frac{2}{7}x+\frac{8}{7}[/tex]
see the attached figure to better understand the problem
Step-by-step explanation:
we have
points L(-3, 6), N(3, 2) and P(1, -8)
Part A) Find the equation of the median from N
we Know that
The median passes through point N to midpoint segment LP
step 1
Find the midpoint segment LP
The formula to calculate the midpoint between two points is equal to
[tex]M(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]
we have
L(-3, 6) and P(1, -8)
substitute the values
[tex]M(\frac{-3+1}{2},\frac{6-8}{2})[/tex]
[tex]M(-1,-1)[/tex]
step 2
Find the slope of the segment NM
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
N(3, 2) and M(-1,-1)
substitute the values
[tex]m=\frac{-1-2}{-1-3}[/tex]
[tex]m=\frac{-3}{-4}[/tex]
[tex]m=\frac{3}{4}[/tex]
step 3
Find the equation of the line in point slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=\frac{3}{4}[/tex]
[tex]point\ N(3, 2)[/tex]
substitute
[tex]y-2=\frac{3}{4}(x-3)[/tex]
step 4
Convert to slope intercept form
Isolate the variable y
[tex]y-2=\frac{3}{4}x-\frac{9}{4}[/tex]
[tex]y=\frac{3}{4}x-\frac{9}{4}+2[/tex]
[tex]y=\frac{3}{4}x-\frac{1}{4}[/tex]
Part B) Find the equation of the right bisector of LP
we Know that
The right bisector is perpendicular to LP and passes through midpoint segment LP
step 1
Find the midpoint segment LP
The formula to calculate the midpoint between two points is equal to
[tex]M(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]
we have
L(-3, 6) and P(1, -8)
substitute the values
[tex]M(\frac{-3+1}{2},\frac{6-8}{2})[/tex]
[tex]M(-1,-1)[/tex]
step 2
Find the slope of the segment LP
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
L(-3, 6) and P(1, -8)
substitute the values
[tex]m=\frac{-8-6}{1+3}[/tex]
[tex]m=\frac{-14}{4}[/tex]
[tex]m=-\frac{14}{4}[/tex]
[tex]m=-\frac{7}{2}[/tex]
step 3
Find the slope of the perpendicular line to segment LP
Remember that
If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)
[tex]m_1*m_2=-1[/tex]
we have
[tex]m_1=-\frac{7}{2}[/tex]
so
[tex]m_2=\frac{2}{7}[/tex]
step 4
Find the equation of the line in point slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=\frac{2}{7}[/tex]
[tex]point\ M(-1,-1)[/tex] ----> midpoint LP
substitute
[tex]y+1=\frac{2}{7}(x+1)[/tex]
step 5
Convert to slope intercept form
Isolate the variable y
[tex]y+1=\frac{2}{7}x+\frac{2}{7}[/tex]
[tex]y=\frac{2}{7}x+\frac{2}{7}-1[/tex]
[tex]y=\frac{2}{7}x-\frac{5}{7}[/tex]
Part C) Find the equation of the altitude from N
we Know that
The altitude is perpendicular to LP and passes through point N
step 1
Find the slope of the segment LP
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
we have
L(-3, 6) and P(1, -8)
substitute the values
[tex]m=\frac{-8-6}{1+3}[/tex]
[tex]m=\frac{-14}{4}[/tex]
[tex]m=-\frac{14}{4}[/tex]
[tex]m=-\frac{7}{2}[/tex]
step 2
Find the slope of the perpendicular line to segment LP
Remember that
If two lines are perpendicular, then their slopes are opposite reciprocal (the product of their slopes is equal to -1)
[tex]m_1*m_2=-1[/tex]
we have
[tex]m_1=-\frac{7}{2}[/tex]
so
[tex]m_2=\frac{2}{7}[/tex]
step 3
Find the equation of the line in point slope form
[tex]y-y1=m(x-x1)[/tex]
we have
[tex]m=\frac{2}{7}[/tex]
[tex]point\ N(3,2)[/tex]
substitute
[tex]y-2=\frac{2}{7}(x-3)[/tex]
step 4
Convert to slope intercept form
Isolate the variable y
[tex]y-2=\frac{2}{7}x-\frac{6}{7}[/tex]
[tex]y=\frac{2}{7}x-\frac{6}{7}+2[/tex]
[tex]y=\frac{2}{7}x+\frac{8}{7}[/tex]
