Name the type of quadrilateral formed, if any, by the following points, and give reasons for
your answer
(-3,5), (1,10),(3,1),(-1,-4)
Answers
Answer:
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.
Types of quadrilaterals and their conditions:
Parallelogram:
In parallelogram both pairs of opposite sides are equal.Diagonals bisect each other.
Rectangle:
In rectangle diagonals are equal.Diagonal bisect each other.Opposite sides are equal.
Rhombus:
In Rhombus all the sides are equal.Diagonals bisect each other at right angle.Diagonals are not equal.
Square:
In square all sides are equal.Diagonals are equal.Diagonals bisect each other at right angle.