MO8-Z8-I-5 2017

Identical rectangles ABCD and EFGH are positioned such that their sides are parallel to the same. The points I, J, K, L, M and N are the intersections of the extended sides, as shown. The area of the BNHM rectangle is 12 cm², the rectangle MBCK area is 63 cm² and the rectangle MLGH area is 28 cm². Find the area of the IFJD rectangle.

Correct result:

S =  418 cm²

Solution:

$$\begin{gathered} S_1=12{\text{cm}}^2 \\ {S}_2=63-S_1=63-12=51{\text{cm}}^2 \\ {S}_3=28-S_1=28-12=16{\text{cm}}^2 \\ a=\mathrm{\mid }MB\mathrm{\mid }=random \\ a=5\text{ cm } \\ {S}_1=a\cdot b \\ {S}_2=a\cdot c \\ {S}_3=b\cdot d \\ b=S_1\mathrm{/}a=12\mathrm{/}5={\displaystyle \frac{12}{5}}=2.4\text{ cm } \\ c=S_2\mathrm{/}a=51\mathrm{/}5={\displaystyle \frac{51}{5}}=10.2\text{ cm } \\ d=S_3\mathrm{/}b=16\mathrm{/}2.4={\displaystyle \frac{20}{3}}\doteq 6.6667\text{ cm } \\ x=d+a+d=6.6667+5+6.6667={\displaystyle \frac{55}{3}}\doteq 18.3333\text{ cm } \\ y=c+b+c=10.2+2.4+10.2={\displaystyle \frac{114}{5}}=22.8\text{ cm } \\ S=x\cdot y=18.3333\cdot 22.8=418{\text{cm}}^2 \end{gathered}$$

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