Name the type of quadrilateral formed by the points A(9,0) B(-9,6) and C( -9,0) using distance formula and give reason for your answer..
Answers
Answer:
Solution-
We shall use the distance formula to calculate the lengths of AB,BC,CD & DA.
AB=
(5−1)
2
+(4−2)
2
=
20
units.
BC=
(5−3)
2
+(4−8)
2
=
20
units.
CD=
(5−1)
2
+(4−2)
2
=
20
units.
DA=
(3+1)
2
+(8−6)
2
=
20
units.
So, AB=BC=CD=DA.
∴ABCD is either a rhombus or a square.
Now, we calculate the slopes of each line.
m
AB
=
5−1
4−2
=
2
1
,
m
BC
=
3−5
8−4
=−2,
m
CD
=
3+1
8−6
=
2
1
,
m
DA
=
1+1
2−6
=−2,
∴m
AB
×m
BC
=
2
1
×(−2)=−1
⇒∠ABC=90
o
.
m
BC
C×m
CD
=(−2)×
2
1
=−1
⇒∠BCD=90
o
.
m
CD
C×m
DA
=
2
1
×(−2)=−1
⇒∠CDA=90
o
.
So, all the angles are 90
o
and the sides are equal.
∴ABCD is a square (also it is a rhombus).
∴ar.ABCD=(AB)
2
=(
20
)
2
sq.units=20 sq. units.
We know that the figure, made by joining the mid points of a square, has an area = half of the square.
∴A(□PQRS)=
2
1
×A(□ABCD)=
2
1
×20=10 sq. units.
So, ABCD is a square (also it is a rhombus) and A(□PQRS)=10 units.
Step-by-step explanation:
AB = √(9-9)²+ (6-0)²
=6
BC = √(9-9)²+ (6-6)²
=18
CD = √(9-9)²+ (0-6)²
= 6
DA = √(9-9)²+ (0-0)²
= 18