Given a triangle whose sides are 24 cm, 30 cm, and 36 cm. Find the radius of a circle which is tangent to the shortest and longest side of the triangle and whose center lies on the third side?

1 Answer
Jan 29, 2018

r=(9sqrt7)/2~~11.91 cm

Explanation:

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Given : AB=24, BC=30 and AC=36,
Let D and E be the tangancy point on AB and AC, respectively, and let O and r be the center and the radius of the circle, respectively, as shown in the figure.
Use Heron's formula to find the area of DeltaABC,
A_(DeltaABC)=sqrt(s(s-a)(s-b)(s-c)
where a,b and c are the side lengths of the triangle, and
s=(a+b+c)/2
=> s=(24+30+36)/2=45
=> A_(DeltaABC)=sqrt(45(45-24)(45-30)(45-36))=sqrt(45*21*15*9)=135sqrt7 " cm^2
Now, A_(DeltaAOB)=1/2*AB*r=1/2*24*r=12r
A_(DeltaAOC)=1/2*AC*r=1/2*36*r=18r
=> A_(DeltaAOB)+A_(DeltaAOC)=A_(DeltaABC)
=> 12r+18r=135sqrt7
=> 30r=135sqrt7
=> r=(135sqrt7)/30=(9sqrt7)/2~~11.91 cm