Z8–I–5 MO 2019

For eight different points as shown in the figure, points C, D, and E lie on a line parallel to line AB, F is the midpoint of line AD, G is the midpoint of line AC, and H is the intersection of lines AC and BE. The area of ​​triangle BCG is 12 cm² and the area of ​​quadrilateral DFHG is 8 cm²

Determine the areas of triangles AFE, AHF, ABG, and BGH.

Correct answer:

S₁ =  12 cm²
S₂ =  12 cm²
S₃ =  4 cm²
S₄ =  4 cm²

Step-by-step explanation:

$$\begin{gathered} S(BCG)=12{\text{cm}}^2 \\ S(DFHG)=8{\text{cm}}^2 \\ S=\frac{av}{2} \\ {S}_1=S(AFE)=S(BCG) \\ {S}_1=12=12{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} S_2=S(ABG)=S(BCG)=S_1 \\ {S}_2=S_1=12=12{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} S(ABD)=2\cdot S(BCG)=2\cdot 12=24 \\ {S}_3=S(AHF)=S(ABD)-S(DFHG)-S(ABG) \\ {S}_3=24-8-12=4{\text{cm}}^2 \end{gathered}$$

$$\begin{gathered} S_4=S(BGH)=S(AHF) \\ {S}_4=S_3=4=4{\text{cm}}^2 \end{gathered}$$

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