Two opposite vertices of a square are (3,4) and (-1,1) .Find the coordinates of the two vertices .
Answers
Answer:
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Answer:
Let A(3,4),B(x,y),C(1,−1) and D be the vertices of the square.
For a square, diagonal = √2×side
AC=√2×AB
[(1−3)
2
+(−1−4)
2
]
=
2
×
[(x−3)
2
+(y−4)
2
]
Squaring both sides,
[(1−3)
2
+(−1−4)
2
]=2×[(x−3)
2
+(y−4)
2
−2
2
+(−5)
2
=2[(x−3)
2
+(y−4)
2
]
4+25=2[(x−3)
2
+(y−4)
2
]
29=2[(x−3)
2
+(y−4)
2
]
[(x−3)
2
+(y−4)
2
]=29/2 ..(i)
AB=BC [sides of a square]
[(x−3)
2
+(y−4)
2
]
=
[(x−1)
2
+(y+1)
2
]
Squaring both sides,
[(x−3)
2
+(y−4)
2
]=[(x−1)
2
+(y+1)
2
]
x
2
−6x+9+y
2
−8y+16=x
2
−2x+1+y
2
+2y+1
−6x+2x−8y−2y=1+1−9−10
−4x−10y=−23
4x+10y=23
y=(23−4x)/10 ...(ii)
Substitute y in (i)
[(x−3)
2
+((23−4x)/10)−4)
2
]=29/2
2
29
=(x−3)
2
+((
10
23−4x
)−4)
2
2
29
=(x
2
−6x+9)+(
10
23−4x−40
)
2
2
29
=100(
100
x
2
−6x+9
)+
100
(−17−4x)
2
2
2900
=(100x
2
−600x+900+16x
2
+136x+289)
1450=116x
2
−464x+1189
116x
2
−464x−261=0
x=
2×116
464±
464
2
−4×116×−261
x=
232
464±
215296+121104
x=
232
464±
33600
x=
232
464±
33600
x=
232
464+580
orx=
232
464−580
x=
232
1044
orx=
232
−116
x=
2
9
orx=
2
−1
x=
2
9
orx=
2
−1