+0  
 
+1
60
3
avatar+86 

In triangle ABC, AB = 3, AC = 5, and BC = 4. The medians AD, BE, and CF of triangle ABC intersect at the centroid G. Let the projections of G onto BC, AC, and AB be P, Q, and R, respectively. Find GP + GQ + GR.

MIRB15  Jul 14, 2017
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3+0 Answers

 #1
avatar+11635 
0

DELETED

rosala  Jul 14, 2017
edited by rosala  Jul 14, 2017
 #2
avatar+75017 
+1

 

The intersection of the medians divide  triangle ABC into 3 triangles with equal areas, triangles AGB, BGC and  AGC

 

And since the area of the large triangle is 6, the areas of each of these triangles is 2

 

So.....

 

Area of AGB  =  (`1/2)  (AB ) (GR)  = 2

 (`1/2)  (3 ) (GR)  = 2

(3) GR  = 4

GR = 4/3

 

Area of BGC  =  (`1/2)  (BC ) (GP)  = 2

 (`1/2)  (4 ) (GP)  = 2

(4) GP  = 4

GP = 1

 

Area of AGC  =  (1/2)  (AC ) (GQ)  = 2

 (`1/2)  (5 ) (GQ)  = 2

(5) GQ  = 4

GQ = 4/5

 

So.....GP + GQ + GR  =  1 + 4/5 +  4/3  =   47 / 15

 

 

 

cool cool cool

CPhill  Jul 14, 2017
edited by CPhill  Jul 14, 2017
 #3
avatar+45 
0

DELETED

arifh556  Jul 14, 2017
edited by arifh556  Jul 14, 2017

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