Question : In a triangle, if three altitudes are equal, then the triangle is:
Option 1: obtuse
Option 2: equilateral
Option 3: right-angled
Option 4: Isosceles
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Correct Answer: equilateral
Solution :
AD, BE and CF are the altitudes drawn on sides BC, CA and AB of $\triangle$ABC such that AD = BE = CF.
Since area of triangle = $\frac{1}{2}$ × base × corresponding altitude,
Area of $\triangle$ABC = $\frac{1}{2}$ × BC × AD = $\frac{1}{2}$ × AB × CF = $\frac{1}{2}$ × CA × BE
⇒ BC × AD = AB × CF = CA × BE
Since AD = BE = CF,
⇒ BC = AB = CA
⇒ $\triangle$ABC is an equilateral triangle.
Hence, the correct answer is equilateral.
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