Step-by-step explanation: (i) Let the given points are A(−1,−2), B(1,0), C(−1,2)and D(−3,0) Then, AB= (1+1) 2 +(0+2) 2 = 2 2 +2 2 = 4+4 = 8 BC= (−1−1) 2 +(2−0) 2 = (2 2 +2 2 ) = 4+4 = 8 CD= ((−3)−(−1)) 2 +(0−2) 2 = 2 2 +(−2) 2 = 4+4 = 8 DA= (−3)−(−1)) 2 +(0−(−2)) 2 = (−2) 2 +2 2 = 4+4 = 8 AC ((−1)−(−1)) 2 +(2−(−2)) 2 = 0+4 2 = 16 =4 BD= (−3−1) 2 +(0−0) 2 = −4 2 = 16 =4 Since the four sides AB,BC,CD and DA are equal and the diagonals AC and BD are equal . ∴ Quadrilateral ABCD is a square. (ii)Let the given points are A(−3,5),B(3,1),C(0,3) and D(−1,−4)Then AB= (−3−3) 2 +(5−1) 2 = (−6) 2 +4 2 = 36+16 = 52 BC= (3−0) 2 +(1−3) 2 = (3 2 +(−2)2 2 ) = 9+4 = 11 CD= (0−(−1)) 2 +(3−(−4)) 2 = 1 2 +(7) 2 = 1+49 = 50 DA= (−1)−(−3)) 2 +((−4)−5)) 2 = (2) 2 +(−9) 2 = 4+81 = 85 Here AB =BC =CD =DA ∴ it is a quadrilateral. (iii)Let the given points are A(4,5),B(7,6),C(4,3) and D(1,2)Then AB= (7−4) 2 +(6−5) 2 = 3 2 +1 2 = 9+1 = 10 BC= (4−7) 2 +(3−6) 2 = ((−3) 2 +(−3) 2 ) = 9+9 = 18 CD= (1−4) 2 +(2−3) 2 = (−3) 2 +(−1) 2 = 9+1 = 10 DA= (1−4) 2 +(2−5) 2 = (−3) 2 +(−3) 2 = 9+9 = 18 AC (4−4) 2 +(3−5) 2 = 0+(−2) 2 = 4 =2 BD= (1−7) 2 +(2−6) 2 = (−6) 2 +(−4) 2 = 36+16 = 52 Here AB=CD,BC=DA . But AC =BD Hence the pairs of opposite sides are equal but diagonal are not equal so it is a parallelogram. hope it helps pls mark as brainliest Reply
Step-by-step explanation:
(i) Let the given points are A(−1,−2), B(1,0), C(−1,2)and D(−3,0) Then,
AB=
(1+1)
2
+(0+2)
2
=
2
2
+2
2
=
4+4
=
8
BC=
(−1−1)
2
+(2−0)
2
=
(2
2
+2
2
)
=
4+4
=
8
CD=
((−3)−(−1))
2
+(0−2)
2
=
2
2
+(−2)
2
=
4+4
=
8
DA=
(−3)−(−1))
2
+(0−(−2))
2
=
(−2)
2
+2
2
=
4+4
=
8
AC
((−1)−(−1))
2
+(2−(−2))
2
=
0+4
2
=
16
=4
BD=
(−3−1)
2
+(0−0)
2
=
−4
2
=
16
=4
Since the four sides AB,BC,CD and DA are equal and the diagonals AC and BD are equal .
∴ Quadrilateral ABCD is a square.
(ii)Let the given points are A(−3,5),B(3,1),C(0,3) and D(−1,−4)Then
AB=
(−3−3)
2
+(5−1)
2
=
(−6)
2
+4
2
=
36+16
=
52
BC=
(3−0)
2
+(1−3)
2
=
(3
2
+(−2)2
2
)
=
9+4
=
11
CD=
(0−(−1))
2
+(3−(−4))
2
=
1
2
+(7)
2
=
1+49
=
50
DA=
(−1)−(−3))
2
+((−4)−5))
2
=
(2)
2
+(−9)
2
=
4+81
=
85
Here AB
=BC
=CD
=DA
∴ it is a quadrilateral.
(iii)Let the given points are A(4,5),B(7,6),C(4,3) and D(1,2)Then
AB=
(7−4)
2
+(6−5)
2
=
3
2
+1
2
=
9+1
=
10
BC=
(4−7)
2
+(3−6)
2
=
((−3)
2
+(−3)
2
)
=
9+9
=
18
CD=
(1−4)
2
+(2−3)
2
=
(−3)
2
+(−1)
2
=
9+1
=
10
DA=
(1−4)
2
+(2−5)
2
=
(−3)
2
+(−3)
2
=
9+9
=
18
AC
(4−4)
2
+(3−5)
2
=
0+(−2)
2
=
4
=2
BD=
(1−7)
2
+(2−6)
2
=
(−6)
2
+(−4)
2
=
36+16
=
52
Here AB=CD,BC=DA . But AC
=BD
Hence the pairs of opposite sides are equal but diagonal are not equal so it is a parallelogram.
hope it helps
pls mark as brainliest