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Triangle ABC has an area of 144. Point D is on BC such that BD : DC =1 : 3 and point F is on AB such that AF : FB= 1 : 4. Point E is on AC such that AD, BE, and CF intersect at point G. What is area of triangle GDC?

 Sep 8, 2020
 #1
avatar+126 
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This is a typical AMC 10/12 problem. We apply Mass points. Let C have mass 4. B will then have mass 4*3=12. A will then have mass 12*4=48. So F will have mass 12+48=60. So FG:GC=1:15. The height of GDC is 15/16 the large triangle, and the base is 4/5 the large triangle. Our answer is 144*15/16*4/5=108

 

smiley

 Sep 8, 2020
 #2
avatar+176 
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Let Δ ABC be the right triangle with AB = 18 and BC = 16.         18 * 16 / 2 = 144 u2

 

∠ADB = arctan(AB / BD) 77.471º

 

∠GDC = 180 - 77.471 = 102.529º

 

∠BCF = arctan(BF / BC) = 41.987º

 

3 more steps:

 

1/   Use the law of sines to calculate the length of a side GC

 

2/   Use trigonometry to calculate the height from D to GC

 

3/   Multiply side GC by the height and then divide by 2

 

Have fun!  smiley     (btw, the answer is not 108) 

 Sep 8, 2020

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