The correct options are
A Square
B 10 sq. unitsSolution-
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.
mAB=4−25−1=12,
mBC=8−43−5=−2,
mCD=8−63+1=12,
mDA=2−61+1=−2,
∴mAB×mBC=12×(−2)=−1
⇒∠ABC=90o.
mBCC×mCD=(−2)×12=−1
⇒∠BCD=90o.
mCDC×mDA=12×(−2)=−1
⇒∠CDA=90o.
So, all the angles are 90o and the sides are equal.
∴ABCD is a square (also it is a rhombus).
∴ar.ABCD=(AB)2=(√20)2sq.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)=12×A(□ABCD)=12×20=10 sq. units.
So, ABCD is a square (also it is a rhombus) and A(□PQRS)=10 units.