Incorrect Answer For Mini-Question?
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For the mini question, shouldn't the answer be
$$r={\sqrt{(s-a)(s-b)(s-c)}\over s} $$instead of
$$r={\sqrt{(s-a)(s-b)(s-c)\over s}} $$The denominator shouldn't be inside the square root, right?
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@aaron-wang Thanks for asking! Actually, the area formula equals \( \sqrt{ s(s-a)(s-b)(s-c)},\) which I can write like this: \(\sqrt{s}\sqrt{(s-a)(s-b)(s-c)}.\) So since the area of the triangle is equal to \(rs,\) dividing this by \(s\) makes the \(\sqrt{s}\) cancel out but leaves a \(\sqrt{s}\) on the bottom.
$$\begin{aligned} r &= \frac{\textcolor{red}{\sqrt{s}} \sqrt{(s-a)(s-b)(s-c)}}{\textcolor{red}{s}} \\\\ &= \frac{\textcolor{red}{\sqrt{s}} \sqrt{(s-a)(s-b)(s-c)}}{\textcolor{red}{\sqrt{s}\sqrt{s}}} \\\\ &= \frac{\sqrt{(s-a)(s-b)(s-c)} }{\sqrt{s}}\\ \end{aligned} $$ -
@debbie Ah, ok! So since the denominator is
$$\sqrt{s} $$it gets combined into that bigger square root.
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