How can I find an equation for the perpendicular bisector of the line segment that has the endpoints , (9,7) and (−3,−5)?

goldenlink7ydw

goldenlink7ydw

Answered question

2023-02-17

How can I find an equation for the perpendicular bisector of the line segment that has the endpoints , (9,7) and (−3,−5)?

Answer & Explanation

Haylie Long

Haylie Long

Beginner2023-02-18Added 7 answers

(9,7) and (-3,-5)
We start by locating the intersection of the segment that connects these two points. These are the midpoint formulas:
Midpoint ( x 1 + x 2 2 , y 1 + y 2 2 ) = ( 9 - 3 2 , 7 - 5 2 )
Midpoint (3,1)
The slope of the line segment is then determined:
s l o p e = m 1 = ( y 2 - y 1 x 2 - x 1 ) = ( - 5 - 7 - 3 - 9 ) = - 12 - 12 = 1
The slope m 2 of a line that is perpendicular to this line can be calculated using the equation:
m 1 m 2 = - 1
( 1 ) ( m 2 ) = - 1
m 2 = - 1
Now, we can write the equation of the perpendicular bisector. We know its slope is #-1# and it goes through the midpoint #(3,1)#.
The equation of a straight line in slope-intercept form is:
y = m x + b where m is the slope and b is the y-intercept.
y = m 2 x + b
y = - x + b
We can use the coordinates of the midpoint in this equation to solve for b:
1 = - 3 + b
b=4
Therefore, the equation of the perpendicular bisector is:
y = - x + 4
Warren Velez

Warren Velez

Beginner2023-02-19Added 1 answers

The midpoint of a line segment with endpoints, ( x 1 , y 1 ) and ( x 2 , y 2 ) is:
( x mid , y mid ) = ( x 1 + x 2 2 , y 1 + y 2 2 )
Substitute the given points:
( x mid , y mid ) = ( 9 + - 3 2 , 7 + - 5 2 )
( x mid , y mid ) = ( 6 2 , 2 2 )
( x mid , y mid ) = ( 3 , 1 )
Please note that the point must be traversed by the perpendicular bisector (3,1).
Calculate the line segment's slope:
m = y 1 - y 2 x 1 - x 2
Substitute the given points:
m = 7 - - 5 9 - - 3
m = 12 12
m = 1
The slope, n, of a line perpendicular to the line segment is:
n = - 1 m
Substitute the value of m:
n = - 1 1
n = - 1
Use the point-slope form of the equation of a line:
y = n ( x - x mid ) + y mid
Substitute the slope, n, and the midpoint:
y = - 1 ( x - 3 ) + 1
We simplify and find that the equation of the perpendicular bisector is
y = 4 - x

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