Question

Say in(Triangle) ◇ABC ,D,E,F are three points on
BC,CA,AB respectively, such that AE:EC=1:3, DC:DB=2:1, BF:FA=3:2

What is the area of (Triangle) ◇HIG?

Answer 1

Answer on Question #72600, Math / Geometry

Say in (Triangle) $\Delta ABCD,E,F$ are three points on $BC,CA,AB$ respectively, such that $AE:EC = 1:3,DC:DB = 2:1,BF:FA = 3:2$ .

What is the area of (Triangle) $\Delta DEF$ ?

Solution

$$S _ {\Delta A B C} = \frac {1}{2} (A B) (A C) \sin A = \frac {1}{2} (A B) (B C) \sin B = \frac {1}{2} (A C) (B C) \sin C$$ $$A E: E C = 1: 3 = > A E = \frac {1}{4} A C, E C = \frac {3}{4} A C$$ $$D C: D B = 2: 1 = > D C = \frac {2}{3} B C, D B = \frac {1}{3} B C$$ $$B F: F A = 3: 2 = > B F = \frac {3}{5} A B, F A = \frac {2}{5} A B$$ $$S _ {\Delta A B C} = S _ {\Delta A F E} + S _ {\Delta F B D} + S _ {\Delta D C E} + S _ {\Delta D E F}$$ $$S _ {\Delta A F E} = \frac {1}{2} (F A) (A E) \sin A = \frac {1}{2} \left(\frac {2}{5} A B\right) \left(\frac {1}{4} A C\right) \sin A = \frac {1}{1 0} S _ {\Delta A B C}$$ $$S _ {\Delta F B D} = \frac {1}{2} (B F) (D B) \sin B = \frac {1}{2} \left(\frac {3}{5} A B\right) \left(\frac {1}{3} B C\right) \sin B = \frac {1}{5} S _ {\Delta A B C}$$ $$S _ {\Delta D C E} = \frac {1}{2} (D C) (E C) \sin C = \frac {1}{2} \left(\frac {2}{3} B C\right) \left(\frac {3}{4} A C\right) \sin B = \frac {1}{2} S _ {\Delta A B C}$$ $$S _ {\Delta D E F} = S _ {\Delta A B C} - \frac {1}{1 0} S _ {\Delta A B C} - \frac {1}{5} S _ {\Delta A B C} - \frac {1}{2} S _ {\Delta A B C} = \frac {1}{5} S _ {\Delta A B C}$$

Answer: $S_{\Delta DEF} = \frac{1}{5} S_{\Delta ABC}$ .

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