As shown in the figure, it is known that the side length of square ABCD is 2, e is a point on DC, and after clockwise rotation, △ADE coincides with △ABF.

(1) indicates the center and angle of rotation;

(2) If EF is connected, what kind of triangle is △AEF? Please explain the reason.

(3) It is known that point G is on BC, and ∠ GAE = 45.

① try to explain ge = de+BG.

② If e is the midpoint of DC, find the length of BG.

Answer:

(1) The rotation center is point A and the rotation angle is 90.

(2) An isosceles right triangle. Reason:

According to the nature of rotation, AF=AE, ∠ FAE = 90, so △FAE is an isosceles right triangle.

(3) ① ∠ GAE = 45, ∠ FAE = 90, then ∠ GAF = 45, that is? ∠GAF=∠GAE

According to the nature of rotation, AF=AE.

And AG=AG.

So △ DAF △ DAE (SAS)?

So FG=GE, which means GE=FB+BG?

②? E is the midpoint of DC, so DE=EC=FB= 1.

Let GB=x, then GC=2-x and Ge =1+X.

∠ c = 90 in Rt△ECG, which is obtained from Pythagorean theorem.

1+(2-x)2=( 1+x)2

Solve this equation to get x=2/3, which means BG=2/3?