Problem 75

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Problem 75: (2017 AMC 10b, 19; 2017 AMC12b 15)

Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$? $\textbf{(A)}\ 9:1\qquad$ $\textbf{(B)}\ 16:1\qquad$ $\textbf{(C)}\ 25:1\qquad$ $\textbf{(D)}\ 36:1\qquad$ $\textbf{(E)}\ 37:1$

Solution

Exercise 75: Solution

Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to $B'$ so that $BB'=2\cdot AB$. Similarly extend $\overline{BC}$ beyond $C$ to $C'$ ($CC'=2\cdot BC$), and $\overline{CA}$ beyond $A$ to $A'$ ($AA'=2\cdot CA$). What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?

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