From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. If the lengths of the perpendiculars are a,b,a,b,a,b, and ccc, find the altitude of the triangle.


Answer:

a+b+ca+b+ca+b+c

Step by Step Explanation:
  1. The following figure shows the required triangle:
  2. Let's assume the side of the equilateral triangle ABCABCABC is xxx.
    The area of the triangle ABCABCABC can be calculated using Heron's formula, since all sides of the triangles are known.
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  3. [Math Processing Error]
  4. Similarly, the area of the triangle BOC=ax2BOC=ax2
    and the area of the triangle AOC=cx2AOC=cx2
  5. [Math Processing Error]
  6. By comparing the equations (1)(1) and (2),(2), we get:
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  7. Now, Area(ABC)=34(x)2Area(ABC)=34(x)2
  8. [Math Processing Error]
    By putting the value of xx from the equation (3)(3), we get,
    Altitude of the triangle ABC=32(2(a+b+c)3)ABC=32(2(a+b+c)3)
    Altitude of the triangle ABC=a+b+cABC=a+b+c
  9. Hence, the altitude of the triangle is a+b+ca+b+c.

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