If the perimeter of a triangle ABC is 30 cm, then what is the value of a `cos^(2)(C//2)+ c cos^(2)(A//2)` ?

Options

A. 15 cm
B. 10 cm
C. `(15)/(2) cm`
D. 13 cm

Answer

A

Explanation

We know from properties of triangle that
`"cos"(A)/(2)=sqrt((s(s-a))/(bc))rArr "cos"^(2)(A)/(2)=(s(s-a))/(bc)`
and `"cos"(C)/(2)=sqrt((s(s-c))/(ab))rArr "cos"^(2)(C)/(2)=(s(s-c))/(ab)`
So, `"a cos"^(2)(C)/(2) + "c cos"^(2)(A)/(2)`
`=a((s(s-c))/(ab))+c((s(s-a))/(bc))=(s(s-c+s-a))/(b)`
`=(s(2s-a-c))/(b)=(s(a+b+c-a-c))/(b)=(s.b)/(b)=s`
`=(30)/(2)=15` cm [given that 2s = 30]