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What is the radius of the circle inscribed in triangle ABC if AB = 5, AC=6, BC=7? Express your answer in simplest radical form.

 Aug 10, 2018

What is the radius of the circle inscribed in triangle ABC if AB = 5, AC=6, BC=7? 


Hello friends !


The radius of the circle is r .

\(a=7\\ b=6\\ c=5 \)


\(c^2=a^2+b^2-2ab\cdot cos\gamma\\ \gamma=arccos\frac{a^2+b^2-c^2}{2ab}=arccos\frac{7^2+6^2-5^2}{2\cdot 7\cdot 6}=arccos 0.7142857\\ \gamma=44.4153°\)


\(b^2=a^2+c^2-2ac\ cos\beta\\ \beta=arccos\frac{a^2+c^2-b^2}{2ac}=arccos\frac{7^2+5^2-6^2}{2\cdot 7\cdot 5}=arccos\ 0.54285714\\ \beta=57.12165°\)


\(a=s+t\)   (two sections on the side a,

                     on both sides of the contact with the circle)

\(tan\frac{\gamma}{2}=\frac{r}{s}\\ s=\frac{r}{tan \frac{\gamma}{2}}\)             \(tan \frac{\beta}{2}=\frac{r}{t}\\ t= \frac{r}{tan \frac{\beta}{2}}\)

\(a=\frac{r}{tan \frac{\gamma}{2}}+\frac{r}{tan \frac{\beta}{2}}=7\)

\(\frac{r}{tan \frac{44.4153°}{2}}+\frac{r}{tan \frac{57.12165°}{2}}=7\)


\(0.544331r+0.408248r=7\cdot 0.408248\cdot 0.544331\\ r=\frac{7\ \cdot\ 0.408248\ \cdot\ 0.544331}{0.544331+0.408248}\)




\(The\ radius\ of\ the\ circle\ inscribed\ in\ triangle\ ABC\ is\ 1.63299.\)


laugh  !

 Aug 10, 2018
edited by asinus  Aug 10, 2018

First, we illustrate the problem:

I - the incenter (center of inscribed circle)

Then, draw lines like this:

Notice that the triangle has been split into 3 smaller ones, each with a height of the radius, and with a base of the sides of the large triangle. Which means that we can say:


\(a\triangle ABC=a\triangle AIB+a\triangle BIC+a\triangle CIA\)

\(a\triangle ABC={5r\over 2}+{6r\over 2}+{7r\over 2}\)

\(a\triangle ABC =9r\)


Now, we also know that \(a\triangle ABC=\sqrt{s(s-AB)(s-AC)(s-BC)}\) wherein \(s={AB+AC+BC\over2}\). This is called Heron's Formula. Substituting the values, we get:






 Aug 10, 2018

Nice approach, Mathehemathh    !!!!


cool cool cool

CPhill  Aug 10, 2018

Herons Formula, when so applied, is something completely new to me.
Thank you, Mathhhemathh!

laugh  !

 Aug 10, 2018

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