數學問題{Deductive Geometry(II)}

1a)
因為AAA
∴△ABC and△ADE are congruent
AB/BC=AD/DE(corr.sides,～△)
12/x=(12 4)/12
x=9

AB/AC=AD/AE(corr.sides,～△)
12/y=(12 4)/(y 6)
y=18

------------------------------------------------
我先假設番AC⊥BD,亦即係BCD係一條直線
∠ABC=∠DEC(given)
∠ACB=∠DCE(given)
∠BAC=180˚-∠ABC-∠ACB(adj. ∠s on st. line)
=180˚-∠DEC-∠DCE
=∠EDC
∴△ABC nad △DEC are congruent.
AB/BC=DE/EC(corr.sides,～△)
20/12=10/x
x=6
AC=√(AB^2-BC^2)(pyth. theorem)
=16
AB/AC=DE/DC(corr.sides,～△)
20/16=10/y
y=8

2008-07-08 16:13:04 補充：
回應eelyw:
[intercept theorem]只能在兩條邊一樣的情況下才能使用

a)By intercept theorem. AC/AB= CE/BD, that is y/12 = 6/4, therefore, y = 12 x 6/4
= 18.
Triangle ABC is similar to triangle ADE, therefore, 12/x = (12 +4)/12 = 16/12 = 4/3.
Therefore, x = 9.
b) Triangle ABC is similar to triangle CDE, therefore, AB/BC = ED/CE
20/12 = 10/x, therefore, x=6.
By Pythagoras theorem, y^2 = 10^2 - x^2 = 100 - 36 = 64. Therefore, y=8.