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 answer:

S =  418 cm²

Step-by-step explanation:

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

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