16. As shown in the figure, fold the rectangular piece of paper ABCD with the crease MN. Points M and N are on the sides AD and BC respectively. The corresponding points of points C and D are points E and F respectively, and point F is inside the rectangle. The extension line of

16. As shown in the figure, fold the rectangular piece of paper ABCD with the crease MN. Points M and N are on the sides AD and BC respectively. The corresponding points of points C and D are points E and F respectively, and point F is inside the rectangle. The extension line of MF intersects BC at point G, and EF intersects BC at point H. EN=2, AB=4. When H is the trisection point of GN, the length of MD is ________

4. (1) As shown in the figure 1. ABO and COD are isosceles right triangle , ∠AOB=∠COD=90°, OA=OB, OC=OD, point C is on side OA, point D is on the extension line of BO, connecting AD and BC , the quantitative relationship between line segments AD and BC is _____

(2) As shown in Figure 2, rotate COD in Figure 1 clockwise (090°) around point O, (1) Does the conclusion in question still hold? If it is true, please prove your conclusion; if it is not true, please explain the reason;

(3) As shown in Figure 3, if AB=8, point C is a point outside the line segment AB, AC=3√3, connect BC

① If CB is wrapped around the point Rotate C 90° counterclockwise to get CD, connect AD, then the maximum value of AD is ______

② If BC is the hypotenuse of RtBCD (three points B, C, and D are arranged clockwise), ∠CDB=90°, connect AD , when ∠CBD=∠DAB=30°, write the value of AD directly.

Solution: (1) AD=BC

(2) From AOD=BOC, OA=OB, OC=OD, so AODBOC, so AD= BC

(2) ① Taking AC as the right-angled side, construct an isosceles right triangle ACE, AD=BE. According to the melon-bean principle, we can know that the trajectory of point E is a circle,

When B, A, and E are collinear, ADmax=8+3 √6

Comment: The difficulty of this question lies in the expansion of the hand-in-hand model, constructing similar models, and using similarities to solve problems; of course, most students may have considerable difficulty in drawing pictures, and most students cannot solve it!

4. As shown in the figure, in the plane rectangular coordinate system , the image of the parabola y=ax+bx-3 passes through points B(6,0), D(4,-3), and the other intersection point with the x axis is A. , intersects the y axis at point C, draw a straight line AD.

(1) ① Find the expression of the parabola ; ② Directly write the function expression of the straight line AD;

(2) Point D is the parabola below the straight line AD At the previous point, connect BE and AD at point F, connect BD, DE, and the area of ​​BDF is recorded as S1, and the area of ​​DEF is recorded as S2. When S1=2S2, find the coordinates of point E;

(3) Point G is a parabola At the vertex of the parabola image, fold the part below the x axis upward along the x axis, and form a new curve with the remaining part of the parabola, marked C1, the corresponding point of the recorded point is C′, and the corresponding point of point G is G ′, translate the curve C1 downward along the y axis n (0n) unit lengths. Among the common points of the curve C1 and the straight line BC, select two common points as P and Q. If the quadrilateral C′G′QP is an parallelogram , directly write the coordinates of point P.

Comments: The question not only tests the folding of the quadratic function image, but also tests the translation, the analytical expression of the function, the existence of a parallelogram, and the calculation, all of which are difficult. The classmates are a huge test.