Answer:
Given endpoint are (4,1) and (2, -5).
For any two points [tex](x_1 , y_1)[/tex] and [tex](x_2 , y_2)[/tex]
Slope of the line is given by:
Slope(m) = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Slope of the segment for the given end points are:
[tex]m_1= \frac{-5-1}{4-2} =\frac{-6}{2} = -3[/tex]
Now, to find the midpoint of line segment.
Midpoint is halfway between the two end points.
then its y value is halfway between the two y values and Its x value is halfway between the two x values.
i.e,
Midpoint = [tex](\frac{x_1+x_2}{2} , \frac{y_1+y_2}{2} )[/tex]
Midpoint of the given line segment is;
[tex](\frac{4+2}{2} , \frac{1-5}{2} ) = (\frac{6}{2} , \frac{-4}{2}) = (3, -2)[/tex]
we have to find the equation of line that is perpendicular bisector of the line segment.
Slope for the perpendicular bisector [tex]m_2[/tex] ;
[tex]m_1 \times m_2 = -1[/tex]
[tex]-3 \times m_2 = -1[/tex]
⇒[tex]m_2 = \frac{1}{3}[/tex]
Point slope form: An equation of a straight line in the form [tex]y-y_1 = m(x -x_1)[/tex] where m is the slope of the line and [tex](x_1, y_1)[/tex] are the coordinates of a given points on the line
Using point slope form to find the equation of line that is the perpendicular bisector;
[tex]y - (-2) = \frac{1}{3}(x-3)[/tex]
[tex]y+2 = \frac{1}{3}(x-3)[/tex]
Using distributive property;
[tex]y +2 =\frac{1}{3} x -1[/tex]
or
[tex]y = \frac{1}{3} x -1-2[/tex]
or
[tex]y = \frac{1}{3} x - 3[/tex]
Therefore, the equation of line that is perpendicular bisector of the segment with given end points is; [tex]y = \frac{1}{3} x - 3[/tex]
To indicate the equation of line through (2, -4) and having slope of [tex]\frac{3}{5}[/tex]
Using Point slope form definition :
[tex]y-y_1 = m(x-x_1)[/tex]
then;
[tex]y-(-4)=\frac{3}{5} (x-2)[/tex]
or
[tex]y+4 = \frac{3}{5} (x-2)[/tex]
Using distributive property:
[tex]y+4 = \frac{3}{5}x-\frac{6}{5}[/tex]
Subtract 4 from both sides we get;
[tex]y = \frac{3}{5}x-\frac{6}{5} -4[/tex]
or
[tex]y = \frac{3}{5}x-\frac{26}{5}[/tex]
therefore, the equation of line is , [tex]y = \frac{3}{5}x-\frac{26}{5}[/tex]
Answer:
Step-by-step explanation:
y = mx + b.....the m is for the slope , (2,-4)...x = 2 and y = -4
now we sub
-4 = 3/5(2) + b
-4 = 6/5 + b
-4 - 6/5 = b
-20/5 - 6/5 = b
- 26/5 = b
so ur equation is : y = 3/5x - 26/5 <== slope intercept form
or if u need it in standard form :
y = 3/5x - 26/5....(5)
5y = 3x - 26
-3x + 5y = -26...(-1)
3x - 5y = 26 <==standard form
