A(7,−8) and C(1,4) are vertices of a square ABCD. Find the equation of diagonal BD.
- Ax−2y=−8
- Bx−2y=8
- Cx+2y=−8
- Dx+2y=8
Solution & Step-by-step Explanation
In a square ABCD, the diagonals AC and BD are perpendicular bisectors of each other.
Find the midpoint of AC (which is also the midpoint of BD):
M=(
2
7+1
,
2
−8+4
)=(
2
8
,
2
−4
)=(4,−2)
Find the slope (m
1
) of diagonal AC:
m
1
=
1−7
4−(−8)
=
−6
12
=−2
Since diagonal BD is perpendicular to AC, the slope (m
2
) of BD is:
m
2
=−
m
1
1
=−
−2
1
=
2
1
Find the equation of line BD passing through M(4,−2) with slope
2
1
:
y−(−2)=
2
1
(x−4)
y+2=
2
1
(x−4)
2(y+2)=x−4
2y+4=x−4
x−2y=8
Find the midpoint of AC (which is also the midpoint of BD):
M=(
2
7+1
,
2
−8+4
)=(
2
8
,
2
−4
)=(4,−2)
Find the slope (m
1
) of diagonal AC:
m
1
=
1−7
4−(−8)
=
−6
12
=−2
Since diagonal BD is perpendicular to AC, the slope (m
2
) of BD is:
m
2
=−
m
1
1
=−
−2
1
=
2
1
Find the equation of line BD passing through M(4,−2) with slope
2
1
:
y−(−2)=
2
1
(x−4)
y+2=
2
1
(x−4)
2(y+2)=x−4
2y+4=x−4
x−2y=8