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Triangle ABC has vertices at A(5,8), B(3,-2), and C(6,1). The point D with coordinates (m,n) is chosen inside the triangle so that the three small triangles ABD, ACD and BCD all have equal areas. What is the value of 10m+n?

 Mar 27, 2019
 #1
avatar+111400 
0

The centroid of a triangle will divide the  triangle into  3  equal  areas

 

The coordinates of the  centroid are  given  by

;

 (  [sum of the  x coordinates of  the  vertex points ] / 3 , [ sum of the  y coordinates of the vertex points ] / 3 ) =

 

( [5 + 3 + 6 ]  /  3  , [ 8 + -2 + 1 ] / 3  )  =  (  14/3, 7/3)

 

So

 

10m + n   =

 

10 ( 14/3) + (7/3) =   140/3  +  7/3   = 147/3   =   49

 

 

cool cool cool

 Mar 27, 2019
 #2
avatar+12271 
+1

Triangle ABC has vertices at A(5,8), B(3,-2), and C(6,1). The point D with coordinates (m,n) is chosen inside the triangle so that the three small triangles ABD, ACD and BCD all have equal areas. What is the value of 10m+n?

 

laugh

 Mar 27, 2019

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