Find the equation of the perpendicular bisector of the segment joining the points (2,−6) and (4,0).
- Ax+3y=6
- Bx+3y=−6
- Cx−3y=−6
- Dx−3y=6
Solution & Step-by-step Explanation
Let the given points be A(2,−6) and B(4,0).
Find the midpoint of segment AB, which lies on the perpendicular bisector:
Midpoint M=(
2
x
1
+x
2
,
2
y
1
+y
2
)=(
2
2+4
,
2
−6+0
)=(3,−3)
Find the slope of segment AB (m
1
):
m
1
=
x
2
−x
1
y
2
−y
1
=
4−2
0−(−6)
=
2
6
=3
Find the slope of the perpendicular bisector (m
2
):
Since lines are perpendicular, m
1
×m
2
=−1.
3×m
2
=−1⟹m
2
=−
3
1
Formulate the equation of the line passing through M(3,−3) with slope m
2
=−
3
1
:
y−y
1
=m
2
(x−x
1
)
y−(−3)=−
3
1
(x−3)
3(y+3)=−1(x−3)
3y+9=−x+3
x+3y=3−9
x+3y=−6
Find the midpoint of segment AB, which lies on the perpendicular bisector:
Midpoint M=(
2
x
1
+x
2
,
2
y
1
+y
2
)=(
2
2+4
,
2
−6+0
)=(3,−3)
Find the slope of segment AB (m
1
):
m
1
=
x
2
−x
1
y
2
−y
1
=
4−2
0−(−6)
=
2
6
=3
Find the slope of the perpendicular bisector (m
2
):
Since lines are perpendicular, m
1
×m
2
=−1.
3×m
2
=−1⟹m
2
=−
3
1
Formulate the equation of the line passing through M(3,−3) with slope m
2
=−
3
1
:
y−y
1
=m
2
(x−x
1
)
y−(−3)=−
3
1
(x−3)
3(y+3)=−1(x−3)
3y+9=−x+3
x+3y=3−9
x+3y=−6