Let S be the parallelogram obtained by joining the mid points of the parallelogram T Consider the fo...........

Quantitative Aptitude (CDS) Mensuration 2D

Let S be the parallelogram obtained by joining the mid-points of the parallelogram T. Consider the following statements :

1. The ratio of area of T so that of S is 2 : 1.

2. The perimeter of S is half of the sum of diagonals of T.

Which of the above statements is/are correct?

1 only

2 only

Both 1 and 2

Neither 1 nor 2

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Explanation:

1) Let us draw a parallel line joining opposite vertices of parallelogram c and S. This is parallel to other two sides

Area WXYZ = $\small \frac{1}{2}area(abxz)$
Area (If a triangle & a parallelogram lie on the same base & between same parallel lines then the area of the triangle will be half of the area of parallelogram)

Similarly ,

$\small Area \: xyz = \frac{1}{2}\times area(xcdz)$

Adding above two we get

$\small Area \: wxyz = \frac{1}{2}\times area(abcd)$

Thus statement 1 is true.

2) ab=dc(opposites sides of parallelogram)

1/2 ab = 1/2 dc

wb = yc

Also, ws = sy (diagonals of parallelogram bisects each other)

& wsx = syc ( corresponding angels)

$\small \therefore \Delta wsx\: is \: congurnet\: to\: \Delta syc$

wx = sc

Similarly, yx = sb, zy = as & zw = ds
Adding the above four, we get wx + yz + zy + zw = sc + sb +as +ds
Perimeter of S = ac + bd
Perimeter of S = Sum of diagonal of T
Hence (2) is false.