Consider a rectangle ABCD of area 90 units. The points P and Q trisect AB, and R bisects CD. The diagonal AC intersects the line segments PR and QR at M and N respectively. What is the area of the quadrilateral PQMN?
Explanation:
Let the breadth be 3x and the breadth be y.
3xy = 90 ⇒ xy = 30
V is midpoint of WR. PW || EV ⇒ EV = PW/2
Similarly, FV = WQ/2
∴ EF = PQ/2 = x/2
∆MPA ∼ ∆MEV
Height of ∆MPA with respect to AP: Height (h1)of ∆MEV with respect to EV = AP : EV = x : x/4 = 4 : 1
Let height of ∆MPA = 4k and height (h1) of ∆MEV = k
∴ 4k + k = 5k = y/2
∴ k = y/10
∴ Height (h1) of ∆MEV = y/10
A(△MEV)=12×EV×h1=12×x4×y10=3060=0.375
Similarly, ∆VFN ∼ ∆CRN
Height (h2) of ∆VFN with respect to VF : Height of ∆CRN with respect to CR = VF : CR = x/4 : 3x/2 = 1 : 6
Let height (h2) of ∆VFN = m and height of ∆CRN = 6m
∴ m + 6m = 7m = y/2
∴ m = y/14
∴ Height (h2) of ∆VFN = y/14
A(△VFN)=12×FV×h2=12×x4×y14=30112≈0.27
A(□PQE)=12×(PQ+EF)×y2
=12×x+x2×y2=3xy8=908=11.25 sq.units
A(□PQMN) = A(□PQEF) – A(∆MEV) + A(∆VFN)
= 11.25 – 0.375 + 0.27 ≈ 11.145
Hence, option (d).
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