Equilateral 80851

Kornelia cut off the colored part from the equilateral triangle. The shortest side of the colored triangle is 1/3 the length of the side of the original triangle. Calculate what part of the triangle she cut off.

Correct answer:

x =  2/9

Step-by-step explanation:

$$\begin{gathered} S_1=\frac{\sqrt{3}}{4}\cdot a^2 \\ {a}_2=\frac{1}{3}\cdot a \\ {h}_2=\frac{2}{3}\cdot a\cdot \sin 60° \\ \sin 60°=\surd 3/2 \\ {h}_2=\frac{\sqrt{3}}{3}\cdot a \\ {S}_2=\frac{a_2\cdot h_2}{2}=\frac{\sqrt{3}\cdot a^2}{3\cdot 2\cdot 3}=\frac{\sqrt{3}\cdot a^2}{18} \\ x=S_2:S_1 \\ x=\frac{\frac{\sqrt{3}\cdot a^2}{18}}{\frac{\sqrt{3}}{4}\cdot a^2} \\ x=\frac{\frac{\sqrt{3}}{18}}{\frac{\sqrt{3}}{4}} \\ x=\frac{\frac{1}{18}}{\frac{1}{4}} \\ x=\frac{4}{18}=\frac{2}{9}=0.2222 \end{gathered}$$

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