Question:
The lengths of perpendiculars drawn from any point inside an equilateral triangle are \(a_1\), \(a_2\) and \(a_3\) respectively. Find the length of the side of the equilateral triangle.
The lengths of perpendiculars drawn from any point inside an equilateral triangle are \(a_1\), \(a_2\) and \(a_3\) respectively. Find the length of the side of the equilateral triangle.
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In geometry problems involving equilateral triangles, remember that the sum of perpendiculars from any point inside the triangle to the sides is constant. This property can be useful in finding the side length or other dimensions.
Updated On: Mar 10, 2025
- \( \frac{2}{\sqrt{3}} (a_1 + a_2 + a_3) \)
- \( \frac{1}{3} (a_1 + a_2 + a_3) \)
- \( \frac{1}{\sqrt{3}} (a_1 + a_2 + a_3) \)
- \( \frac{4}{\sqrt{3}} (a_1 + a_2 + a_3) \)
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The Correct Option is A
Solution and Explanation
In an equilateral triangle, the sum of the perpendiculars drawn from any point inside the triangle to the sides of the triangle is constant. Using this property, we can find the side length using the formula provided.
\[
\text{Side length} = \frac{2}{\sqrt{3}} \left( a_1 + a_2 + a_3 \right)
\]
Final Answer: The correct answer is (a) \( \frac{2}{\sqrt{3}} (a_1 + a_2 + a_3) \).
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