Question 21444: pls help me with this one.
Find the ratio between the area of the circle inscribed in a triangle and the area of the circumscribing circle? the answer is 1/4. thank you.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! THE PROBLEM IS NOT CORRECT .IT IS TRUE FOR EQUILATERAL TRIANGLE.
I DO NOT KNOW WHAT YOU ARE STUDYING AND WHAT IS YOUR BACK GROUND KNOWLEDGE..IF IT IS K 12 OR COMPARABLE AND YOU KNOW PROPERTIES OF TRIANGLES UNDER TRIGNOMETRY, I AM GIVING BELOW A PROOF BASED ON THAT.IF THIS IS NOT SUITABLE ,PLEASE COME BACK WITH ANSWERS TO MY ABOVE DOUBTS AND I SHALL GIVE YOU A SUITABLE ALTERNATIVE METHOD COMMENSURATE WITH THAT.
WE USE THE FOLLOWING FORMULAE IN PROPERTIES OF TRIANGLES UNDER TRIGNOMETRY...
RADIUS OF INCIRCLE =r =4R SIN(A/2)*SIN(B/2)*SIN(C/2)..WHERE R=RADIUS OF CIRCUM CIRCLE AND A,B AND C ARE THE ANGLES OF THE TRIANGLE.
FOR EQUILATERAL TRIANGLE A=B=C=60 DEGREES.THEN WE GET
r=4R * SIN (30)*SIN(30)*SIN(30)=4R*(1/2)*(1/2)*(1/2)=R/2
HENCE r/R=1/2
(AREA OF IN CIRCLE)/(AREA OF CIRCUM CIRCLE)= (PI*r*r)/(PI*R*R)
=(r/R)^2=(1/2)^2=1/4...YOU CAN EASILY VERIFY THAT THIS ANSWER WILL NOT BE OBTAINED IF THE TRIANGLE IS NOT EQUILATERAL..IF TRIANGLE HAS A=90,B=60 AND C=30 WE GET RATIO OF AREAS AS 0.1334
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