Question #20606

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

Expert's answer

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)

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)

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