Problem: One of the sides of a triangle is divided into segments of 6 and 8 units by the point of tangency of the inscribed circle. If the radius of the circle is 4, then the length of the shortest side of the triangle is:
Answer Choices:
A. 12 units
B. 13 units
C. 14 units
D. 15 units
E. 16 units
Solution:
Denoting the sides of the triangle by a,b,c we observe that a=8+6=14, b=8+x,c=x+6. ∴2s=a+b+c=2x+28,s=x+14. On the one hand, the area of the triangle is half the product of the perimeter and the radius of the inscribed circle; on the other hand,
it is given in terms of 8 so that Area =r⋅s=4(x+14)=s(s−a)(s−b)(s−c)=48x(x+14) or (x+14)2=3x(x+14),x+14=3x.∴x=7, and the shortest side is c=6+7=13;
