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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.

Guest Aug 10, 2018
 #1
avatar+7623 
+1

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)

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

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

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

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

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

 

\(r=1.63299\)

 

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

 

laugh  !

asinus  Aug 10, 2018
edited by asinus  Aug 10, 2018
 #2
avatar+349 
+2

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:

\(\sqrt{9*4*3*2}=9r\)

\(6\sqrt{6}=9r\)

\(r={2\sqrt{6}\over3}\)

 

laugh

Mathhemathh  Aug 10, 2018
 #3
avatar+92596 
+2

Nice approach, Mathehemathh    !!!!

 

cool cool cool

CPhill  Aug 10, 2018
 #4
avatar+7623 
+2

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

laugh  !

asinus  Aug 10, 2018

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