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This is the question that I got in TCS Ninja under the Quantitative section.

Help

How shall I do this? Help !!

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asn
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1 Answers1

2

Denote by $[...]$ the area of the polygon $...$

Lema 1

Let $ABCD$ be a rectangle a $P$ a point inside as shown. It follows that $$[APB]+[DPC]=[APD]+[CPB]$$ enter image description here

Proof

Draw the parallels to $AD$ and $AB$ through $P$. Then $$[APB]+[DPC]=\frac{AB·FP}{2}+\frac{DC·EP}{2}=\frac{AB·FP+AB·DC}{2}=\frac{AB·EF}{2}=\frac{[ABCD]}{2}$$

Now back to your problem $$A1+A3=2081=A2+A4=1016+A4\iff A4=2081-1016=\color{red}{1065}$$

Thanks for the correction @Jean Marie.

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Dr. Mathva
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  • The final calculations should be revised : $A_1+A_3=2081$ etc... – Jean Marie Apr 04 '19 at 15:31
  • You're totally right. Thanks @JeanMarie – Dr. Mathva Apr 04 '19 at 17:55
  • I had a look at this issue trying to find an answer to a contradiction I find in the following question https://math.stackexchange.com/q/3174470 to which I have given an answer some hours ago. Maybe, you have an idea (see my Edit) – Jean Marie Apr 04 '19 at 18:00