Math Help!(1991/II/53)

2010-03-20 9:37 pm

1991/II/53
In the figure (triangle ABC),M is the mid-point of BC and AD=2DB.AM and CD intersects at K.Find area of triangle ADK/area of triangle AKC.
A1/2
B.2/3
C3/4
D4/5
E.1
Would you please show me your steps?

回答 (1)

micatkie

2010-03-20 10:38 pm

✔ 最佳答案

The answer is B.

Let:
X = Area of ΔADK
Y = Area of ΔAKC
U = Area of ΔDKB
V = Area of ΔBKM
W = Area of ΔCKM

Consider X and U:
The two triangles have the same height with bases AD : BC = 2 : 1
X : U = AD : BC = 2 : 1
Hence, U = 0.5X ... (1)

Consider V and W:
The two triangles have the same height with equal bases.
V = W ... (2)

Consider ΔBAM and ΔCAM:
The two triangles have the same height with equal bases.
X + U + V = Y + W ... (3)

Put (1) and (2) into (3):
X + 0.5X + W = Y + W
1.5X = Y
3X = 2Y
X : Y = 2 : 3 .... The answer is B.

參考: micatkie

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