From a point inside an equilateral triangle, three perpendiculars are drawn. length of an arm of the equilateral triangle?

Question

From a point inside an equilateral triangle, three perpendiculars are drawn on its three sides measuring 6cm, 8cm and 12 cm. What is the length of an arm of the equilateral triangle?

Answer 1

If you draw 3 lines from the point to the CORNERS and also add the perpendiculars, using Pythagorus's A^2 + B^2 = C^2 for the 6 inner right triangles formed and using three equations from the segments of the original equilateral triangle, that side 1 = side 2 and side 2 = side 3 and side 1 = side 3, you have 9 equations for your 9 variables and you should be able to solve for the sides. (Note, you can easily change this to 6 variables and 6 equations when you note that the lines from the point to the CORNERS are shared by two triangles each.)

Answer 2

For an "equalateral" triangle there is only one point from which all sides can meet perpendicularly and the distance of these perpendicular lines must be all the same towards each leg; therefore the numbers proposed does not fit boundary of the problem.

 

However, a triangle can be determined from the lengths proposed by observing interior subtriangles with equally spaced angles and applying c^2 = b^2 + a^2 - 2*a*b - cos(ang_a,b)

 

for example, with a=6, b=8 and ang = 120 deg: 

c_1 = sqrt(36+64-96(0.5)) = 7.21;

a=6, b= 12, c_2= 10.39;

a=8, b=12, c_3= 10.58

 

 

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