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triangle

by shashank.ism » Tue Feb 09, 2010 7:13 am
In a triangle ABC, the incircle touches the three sides AB, BC and CA at the points D, E and F respectively. If the length(in cm) of the sides AB, BC and CA are three consecutive even numbers, then which of the following cannot be the radius(in cm) of the incircle?

a 2
b sqrt(7)
c sqrt(15)
d sqrt(32)
e sqrt(3)
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by harsh.champ » Tue Feb 09, 2010 2:18 pm
shashank.ism wrote:In a triangle ABC, the incircle touches the three sides AB, BC and CA at the points D, E and F respectively. If the length(in cm) of the sides AB, BC and CA are three consecutive even numbers, then which of the following cannot be the radius(in cm) of the incircle?

a 2
b sqrt(7)
c sqrt(15)
d sqrt(32)
e sqrt(3)
Since,it is an incircle,all the sides act as tangents.
Tangents from same pt. to a circle are equal.Hence,BD=BE;AD=AF;CF=CE
Also radius is perpendicular to the tangent.
Also,r = A/((a+b+c)/2) where A is the area of the triangle.
s=(a+b+c)/2 which is an integer as a,b,c are even.
Also,A=sqrt[s(s-a)(s-b)(s-c)].Also, all the values are integers.
Hence, r = sqrt(integer)/integer
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