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Mathematics middle school entrance examination questions
A right triangle AOB is known, where ∠ AOB = 90, OA = 2° and OB = 4°. Put the paper in a plane rectangular coordinate system and fold the paper. The crease intersects with the edge OB at point C and with the edge AB at point D. 。

(1) As shown in figure 1, if point B and point O overlap after folding, the coordinate of point D is (1, 2);

(2) As shown in Figure 2, if the folded point B coincides with the point A, find the coordinates of the point C;

(3) As shown in Figure 3, if the point where point B falls on the edge OA after folding is B', let OB'=x and OC=y, try to write the resolution function of Y about X. Test site: Linear function synthesis problem. Analysis: (1) From the center line of CD △OAB, the coordinates of point D can be obtained;

(2) Let OC=m, and from the nature of folding △ ACD △ BCD, then BC=AC=4-m, OA=2. In Rt△AOC, the value of m can be obtained by Pythagorean theorem.

(3) From the nature of folding, we can know that △ b ′ CD △ BCD, if Ob ′ = X and OC=y, then B ′ C = BC = Ob-OC = 4-Y. In RT △ b ′ OC, the functional relationship between Y and X is established by Pythagorean theorem. Solution: solution.

Then CD is the center line of δ△OAB, so d (1, 2).

So the answer is: (1, 2);

(2) As shown in Figure 2, if the folding point B coincides with the point A, then △ ACD △ BCD,

Let the coordinate of point C be (0, m) (m > 0), then BC=OB-OC=4-m, and then AC=BC=4-m,

In Rt△AOC, according to Pythagorean theorem, AC2=OC2+OA2, that is, (4-m)2=m2+22.

The solution is m=32, so c (0 0,32);

(3) As shown in Figure 3, the point where the folding point BB falls on the edge OA is B', then △ b' CD △ BCD,

Let OB'=x and OC=y according to the meaning of the question, then B'C=BC=OB-OC=4-y,

In rt △ b ′ oc, b ′ c2 = oc2+ob ′ 2, that is, (4-y)2=y2+x2, that is, y=- 18x2+2,

Starting from the point B' on the edge OA, there are 0≤x≤2,

Therefore, the resolution function is y =- 18x2+2 (0 ≤ x ≤ 2). Comments: This question examines the comprehensive application of a function. The key is to get congruent triangles from the properties of folding. In a right triangle, the Pythagorean theorem is used to establish the equation, solve the equation or get the functional relationship.