+0  
 
+1
4
1
avatar+209 

In triangle $ABC$, let $I$ be the incenter of triangle $ABC$. The line through $I$ parallel to $BC$ intersects $AB$ and $AC$ at $M$ and $N$, respectively. If $AB = 5$, $AC = 5$, and $BC = 8$, then find the area of triangle $AMN$.

 Apr 17, 2024
 #1
avatar+14943 
+1

 Find the area of triangle AMN.

 

\(\overline{AB} = c=5, \overline{AC} = b=5\ and\ \overline{BC}=a = 8\)

 

\( A = s ( s − a ) ( s − b ) ( s − c )\ Heron's\ formula\\ s=\dfrac{a+b+c}{2}=\dfrac{8+5+5}{2}=9\\ A_{ABC} = 9 ( 9 − 8 ) ( 9 − 5) ( 9 − 5 )\\ A_{ABC}=144\\ Any\ line\ through\ the\ center\ of\ gravity\ of\ a\ triangle\ divides\ it\ into\ equal\ parts.\\A_{AMN=}\frac{1}{2}\cdot A_{ABC}=\frac{1}{2}\cdot 144\\ \color{blue}A_{AMN}=72\)

 

laugh  !

 Apr 17, 2024
edited by asinus  Apr 18, 2024

3 Online Users

avatar
avatar
avatar