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Is it true that the area of the shaded region is 500 $c m 2$ , if, in rectangle ABCD, AB=40cm, BC=25cm and P, Q, R and S are midpoints of AB, BC, CD and DA respectively?

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True

False

answer is A.

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Detailed Solution

We are given that in rectangle ABCD, AB=40cm and BC= 25cm.
Also, P, Q, R and S are midpoints of sides AB, BC, CD and DA respectively, therefore AB = SQ = CD = 40cm and DA = RP = CB = 25cm.
First, we will join these opposite midpoints.
Question Image

We see that SQ and RP bisect each other at point O. This means OR=OP=12.5cm and SO=OQ=20cm.
Now, area of triangle=

12 × b × h

, where b is base and h is height.
In triangle SRQ, SQ=40cm and RO=12.5cm.
Area of triangle SRQ=

12 × S Q × R O

12 × 40 × 12.5

250 c m 2

Likewise in triangle SPQ, SQ=40cm and OP=12.5cm.
Area of triangle SPQ=

12 × S Q × O P

12 × 40 × 12.5

250 c m 2

Area of shaded region= Area of a triangle (SRQ+SPQ)
=

250 c m 2

250 c m 2

=500

c m 2

Therefore, the area of the shaded region is 500

c m 2

.
Hence, the given statement is true and option 1 is correct.

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