the points (-2,-1)(1,0)(x,3)(1,y) form a parallelogram find x and y
Answers
Answer:
et the point of the given parallelogram be A(−2,−1),B(1,0),C(x,3)andD(1,y)
So by distance formula we have,
Distance between two points =
(x
2
−x
1
)
2
+(y
2
−y
1
)
2
AB=
(1+2)
2
+(0+1)
2
=
9+1
=
10
CD
2
=(1−x)
2
+(y−3)
2
=AB
2
=10 (opposite sides)
⇒1+x
2
−2x+y
2
+a−6y=10
⇒x
2
+y
2
−2x−6y=0 ----------(1)
lso, BC
2
=(x−1)
2
+(3)
2
=x
2
+1−2x+9
=x
2
−2x+10
AD
2
=(1+2)
2
+(y+1)
2
=a+y
2
+1+2y
∴BC
2
=AD
2
(opposite sides of a parallelogram)
⇒x
2
+2x+10=9+y
2
+1+2y
⇒x
2
−2x=y
2
+2y ------------(2)
From (1) and (2), we get
y
2
+2y+y
2
−6y=0
⇒2y
2
−4y=0
⇒y
2
−2y=0 ⇒ y(y−2)=0
⇒y=2
x
2
−2x+y
2
−6y=0
⇒x
2
−2x+4−12=0
⇒x
2
−4x+2x−8=0
⇒x(x−4)+2(x−4)=0
⇒(x−4)(x+2)=0
⇒x=4
So answer is x=4 and y=2
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