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A square ABCD, L is the midpoint of side AB, M is the midpoint of side BC, N is the midpoint of side CD and O is the midpoint of side AD, then find the ratio of area of shaded region PQRS to the area of square ABCD.

A square ABCD, L is the midpoint of side AB, M is the midpoint of side BC, N is the midpoint of side CD and O is the midpoint of side AD, then find the ratio of area of shaded region PQRS to the area of square ABCD. Correct Answer 1 ∶ 5

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L and N are the midpoints, so DL ∥ BN.

Similarly, OC ∥ AM,

In ΔAQB, L is the midpoint of AB and LP ∥ BQ.

So, P is a midpoint of side AQ.

⇒ AP= (1/2) × AQ

⇒ AP/AQ = 1/2

Also, ΔAPL and ΔAQB are similar

⇒ Ratio of area of ΔAPL and ΔAQB = 1 ∶ 4

⇒ Area of ΔAPL = 1

⇒ Area of PLQB = 3

By symmetry,

⇒ Area of ΔAPL = ΔQMB = ΔNRC = ΔDOS = 1

⇒ Area of PLQB = RCMQ = DSRN = OSPA = 3

Now, Join mid points O and M,

⇒ Area of ΔAMB = ΔAOM = ΔMOC = ΔDOC = 5

⇒ Area of ABCD = 5 + 5 + 5+ 5 = 20

⇒ Area of AMCO = 5 + 5 = 10

⇒ Area of PQRS = area of AMCO – area of APSO – area of CRQM = 10 – 3 – 3 = 4

∴ Area of shaded region ∶ Area of ABCD = 4 ∶ 20 = 1 ∶ 5

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