Question

If a, b, and c are the sides of a triangle and s is its semi-perimeter, then what is the value of the expression

$\frac{(s-a)+(s-b)+(s-c)}{(\frac{4}{3}s-\frac{5}{4}a)+(\frac{4}{3}s-\frac{5}{4}b)+(\frac{4}{3}s-\frac{5}{4}c)?}$

• A.
  $\frac{1}{3}$
• B.
  0
• C.
  $\frac{2}{3}$
• D.
  $1$

Answer 1

The correct option is C

$\frac{2}{3}$
The sides of the triangle are a, b, and c and s is its semi-perimeter.

$\therefore s=\frac{a+b+c}{2}\Rightarrow a+b+c=2s\dots ..\left(i\right)\therefore \frac{(s-a)+(s-b)+(s-c)}{(\frac{4}{3}s-\frac{5}{4}a)+(\frac{4}{3}s-\frac{5}{4}b)+(\frac{4}{3}s-\frac{5}{4}c)=\frac{3s-(s+b+c)}{3\times \frac{4}{3}s-\frac{5}{4}(a+b+c)}}=\frac{3s-2s}{4s-\frac{5}{4}\times 2s}\left[From\right(i\left)\right]=\frac{s}{4s-\frac{5}{2}s}=\frac{s}{\frac{3}{2}s}=\frac{2}{3}$

Hence, the correct answer is option (c).