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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.

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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

#1
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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)

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

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

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

$$\frac{r}{0.408248}+\frac{r}{0.544331}=7$$

$$0.544331r+0.408248r=7\cdot 0.408248\cdot 0.544331\\ r=\frac{7\ \cdot\ 0.408248\ \cdot\ 0.544331}{0.544331+0.408248}$$

$$r=1.63299$$

$$The\ radius\ of\ the\ circle\ inscribed\ in\ triangle\ ABC\ is\ 1.63299.$$ !

Aug 10, 2018
edited by asinus  Aug 10, 2018
#2
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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:

$$\sqrt{9*4*3*2}=9r$$

$$6\sqrt{6}=9r$$

$$r={2\sqrt{6}\over3}$$ Aug 10, 2018
#3
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Nice approach, Mathehemathh    !!!!   CPhill  Aug 10, 2018
#4
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Herons Formula, when so applied, is something completely new to me.
Thank you, Mathhhemathh! !

Aug 10, 2018