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Mathematics for senior high school entrance examination
analyse

(1) Solve the first problem by using the definition of external angles of triangles, the sum of internal angles of triangles and the properties of isosceles right triangles;

(2) It is proved that △ABD and △DCE are similar, and the functional relationship between Y and X can be obtained by using the similar properties of triangles;

(3) Based on the similarity between △ABD and △DCE, discuss and solve the problem in three situations: AD=AE, AD=DE and AE=DE.

explain

Solution:

( 1)

Guess ∠BDA=∠CED.

Prove:

AB = AC,∠BAC=90

∴∠B=∠C=45

∠∠ADC =∠b+∠ 1 = 45+∠2

∴∠ 1=∠2

∠∠BDA = 180-∠ 1-∠B,∠CED= 180 -∠2-∠C

∴∠ced=∠bda;

(2)

From (1):

∠BDA=∠CED,∠B=∠C

∴△ABD∽△DCE

∴BD/CE=AB/DC

Namely:

x/(4-y)=4/(4√2-x)

∴y=( 1/4)x? -√2x+4(0