A(4,−3) and C(0,7) are vertices of a square ABCD. Find the equation of diagonal BD?
- A2x−5y=6
- B2x+5y=−6
- C2x−5y=−6
- D2x+5y=6
Solution & Step-by-step Explanation
In a square, the diagonals AC and BD are perpendicular to each other and bisect each other.
Step 1: Find the midpoint of AC, which is also the midpoint of BD.
Midpoint M=(
2
4+0
,
2
−3+7
)=(2,2)
Step 2: Find the slope of AC (m
1
).
m
1
=
0−4
7−(−3)
=
−4
10
=−
2
5
Step 3: Find the slope of BD (m
2
). Since BD⊥AC, m
1
×m
2
=−1.
−
2
5
×m
2
=−1⟹m
2
=
5
2
Step 4: Form the equation of line BD passing through M(2,2) with slope
5
2
.
y−2=
5
2
(x−2)
5(y−2)=2(x−2)
5y−10=2x−4
2x−5y=−6
Step 1: Find the midpoint of AC, which is also the midpoint of BD.
Midpoint M=(
2
4+0
,
2
−3+7
)=(2,2)
Step 2: Find the slope of AC (m
1
).
m
1
=
0−4
7−(−3)
=
−4
10
=−
2
5
Step 3: Find the slope of BD (m
2
). Since BD⊥AC, m
1
×m
2
=−1.
−
2
5
×m
2
=−1⟹m
2
=
5
2
Step 4: Form the equation of line BD passing through M(2,2) with slope
5
2
.
y−2=
5
2
(x−2)
5(y−2)=2(x−2)
5y−10=2x−4
2x−5y=−6