Answer to Question #20606 in Geometry for Abhishek

Question

The three perpendicular distances of three sides of an equilateral triangle from a point which lies inside that triangle are 6 cm,9 cm and 12 cm respectively.The perimeter of the triangle is

Answer 1

Triangle ABC
Point O
H1- point of intersection with BC, OH1=12
H2 - point of intersection with AB, OH2=6
BH1=a
angle OBH=å.
12=a*tgå
6/sinå=12/sin(60-å) (sine theorem)
sin(60-å)/sinå=1/2
sin60coså-cos60sinå=1/2sinå
Sqrt(3)/2=tgå
a=24/sqrt(3)
Analogically CH=b
b=12/(2Sqrt(3)5)=30/Sqrt(3))
Side is equal to 54/sqrt(3)
So, perimeter is 54sqrt(3)