Question : $\triangle$ABC is an equilateral triangle in which D, E, and F are the points on sides BC, AC, and AB, respectively, such that AD $\perp$ BC, BE $\perp$ AC and CF $\perp$ AB. Which of the following is true?
Option 1: 4AC$^2$ = 5BE$^2$
Option 2: 3AC$^2$ = 4BE$^2$
Option 3: 2AB$^2$ = 3AD$^2$
Option 4: 7AB$^2$ = 9AD$^2$
Correct Answer: 3AC$^2$ = 4BE$^2$
Solution :
As we know,
Height of the equilateral triangle = $(\frac{\sqrt{3}}{2})$ × (Sides)
Height = AD = BE = CF
Now, $(\frac{\sqrt{3}}{2}$) × AC = BE
⇒ $\sqrt{3}$ AC = 2BE
Squaring on both sides
⇒ 3AC$^2$ = 4BE$^2$
Hence, the correct answer is 3AC$^2$ = 4BE$^2$.
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