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In triangle A, B, C, AB=13, AC=15 and BC=14. Let I be the incenter. The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find the length of BI.

Guest Aug 9, 2018
 #1
avatar+7617 
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In triangle A, B, C, AB=13, AC=15 and BC=14. Let I be the incenter. The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find the length of BI.

 

Hello friends!

 

triangle ABC

a=14   b=15   c=13

 

Referring to the graph in Mathhemathh's reply on August 09, 2018.

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

 

1. circle radius

 

formula of Heron

 

\(A=\sqrt{s(s-a)(s-b)(s-c)}\\ s=\frac{1}{2}(a+b+c)\\ s=\frac{1}{2}(14+15+13)\\ s=21\\ A=\sqrt{21(21-14)(21-15)(21-13)}\)

\(A=84\)

 

\(A=\frac{ar+br+cr}{2}\\ r=\frac{2A}{a+b+c}=\frac{168}{14+15+13}=\frac{168}{42}\\ \color{blue}r=4\)

Thanks Mathhemathh!

 

2. angle \(\beta\)

 

cosine

 

\(b^2=a^2+c^2-2ac\cdot cos \beta\\ \beta=arccos\frac{a^2+c^2-b^2}{2ac}\\ \beta=arccos\frac{14^2+13^2-15^2}{2\cdot 14\cdot 13c}=arccos\ 0.384615\)

\(\beta=67.3801°\) 

\(\frac{\beta}{2}=33.690°\)

 

3. Length of the route \(\overline{BI}\)

 

\(sin\frac{\beta}{2}=\frac{r}{\overline{BI}}\\ \overline{BI}=\frac{r}{sin\frac{\beta}{2}}=\frac{4}{sin\ 33.690°}\)

\(\overline{BI}=7.2111\)

 

\( The\ length\ of\ \overline{BI}\ is\ 7,2111 \)

 

laugh  !

asinus  Aug 11, 2018
edited by asinus  Aug 11, 2018
edited by asinus  Aug 11, 2018
edited by asinus  Aug 11, 2018
 #2
avatar+9726 
+2

In triangle A, B, C, AB=13, AC=15 and BC=14. Let I be the incenter. The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find the length of BI.

laugh

Omi67  Aug 11, 2018

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