D is the midpoint of side BC of ΔABC. Point E lies on AC such that CE = 1/3 AC, BE and AD intersect at G. What is AG/GD?
Show answer & explanation
Correct answer: D
Explanation
The correct answer is option 4 i.e 4 : 1
We drew a line from D to the line AC at point F which is parallel to BE,
In both triangles CEB & CFD, by A-A similarity
ΔBCE ∼ ΔDCF
So,
(BC/DC) = (CE/CF)
⇒ (BD + DC)/DC = (CF + FE)/CF
⇒ 1 + BD/DC = 1 + FE/CF
As we know that BD = DC because D bisects the line BC
⇒ FE = CF
⇒ FE/CE = 1/2 ......... (1)
Similarly in the triangle, ADF & AGE, by A-A similarity
⇒ (AD/AG) = (AF/AE)
⇒ (AG + GD)/AG = (AE + FE)/AE
⇒ 1 + GD/AG = 1 + FE/AE
⇒ AG/GD = AE/FE
From (1) we would get that,
⇒ AG/GD = AE/(CE/2)
⇒ AG/GD = 2 × AE/CE = 2 × 2 : 1
⇒ AG/GD = 4 ∶ 1