If \(r=\) in radius, \(\triangle=\) Area of \(\triangle A B C, s=\) semi-perimeter then which of the…

1. \(\Delta=r+s\)
2. \(\Delta=\frac{r}{S}\)
3. \(\Delta=(r \mathrm{~s})^2\)
4. \(\Delta=r\) s

Solution

Let the length of sides are \(a, b\) and \(c\), so \(s=\frac{a+b+c}{2}\) Now, from the diagram area of triangle \(A B C\) is
\(\begin{aligned} \Delta & =\Delta I B C+\Delta I C A+\Delta I A B \\ & =\frac{1}{2} a r+\frac{1}{2} b r+\frac{1}{2} c r \\ \Delta & =r\left(\frac{a+b+c}{2}\right)=r \end{aligned}\) Hence, option (d) is correct.

Asked in: AP EAMCET 2020 (21 Sep Shift 1)

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