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The vertical and horizontal functions of knowledge (this section mainly refers to linear functions and inverse proportional functions) and the problems of images and geometry are all based on functions to explore geometric properties. It is very important to solve the coordinate problems of several key points by using the properties of functions, so that geometric knowledge and functional knowledge can be organically and naturally combined to break through the difficulties. But when solving this kind of problems, we should pay attention to the relationship between the solution of the equation and the coordinates, and the relationship between the coordinates and the length of the line segment.

Typical example

Example 1 (Taiyuan, Shanxi) As shown in the figure, in the plane rectangular coordinate system, a straight line and a point intersect at one point and one point respectively, and a point is a moving point on a straight line.

(1) Find the coordinates of this point.

(2) When it is an isosceles triangle, find the coordinates of each point.

(3) Is there a point on the line that makes the quadrilateral with this point as its vertex a parallelogram? If it exists, directly write out the value of; If it does not exist, please explain why. Pay attention to solving linear equations and coordinate relations with thinking (1);

(2) When it is an isosceles triangle, it is discussed in three situations.

(3) A quadrilateral with a vertex is a parallelogram.

Three situations?

Example 2 (Huzhou, Zhejiang) is known: in a rectangle, a plane rectangular coordinate system as shown in the figure is established with a straight line as the axis and the axis respectively. It is a moving point (non-coincidence) on the edge, and the image of the inverse proportional function passing through this point intersects with the edge.

(1) Verification: the area is equal to;

(2) Remember, what is the maximum value? What is the maximum value?

(3) Please explore: Is there such a point? After the edge is folded in half, the point just falls on the floor? If it exists, find out the coordinates of the point; If it does not exist, please explain why.

The area of sum is expressed by the algebraic expression of ideas (1); (2) Write the coordinates of two points (including algebraic expressions) and solve them with the triangle area formula; (3) If there is such a point, after the edge is folded in half, this point just falls on the edge, which is proved by passing the point.

Example 3 (Jiaxing, Zhejiang) As shown in the figure, in the rectangular coordinate system, it is known that two points are in the first quadrant and are regular triangles, the positive semi-axis of the intersecting axis of the circumscribed circle is at this point, and the tangent of the circle passing through this point is at this point.

(1) Find the coordinates of two points;

(2) Find the resolution function of the straight line;

(3) Let each be two moving points on the line segment and divide the perimeter of the quadrilateral equally.

Try to explore: the largest area?

The train of thought (1) is;

(2) connect a C to prove CD‖OB. (3) pass.

Geometric figure to establish quadratic function model to solve, pay attention to

What is the range of independent variables?

Example 4 (Hangzhou, 07) is a right-angled trapezoid with a height (as shown in figure 1)? The moving point starts from the point at the same time, the point moves along the point to stop, and the point moves along the point to stop. What is the speed at two o'clock? And the point is up, and the point is just up? Let's start from point at the same time, the elapsed time is, and the area is (as shown in Figure 2)? Take abscissa and ordinate respectively to establish rectangular coordinate system. The function image of the known point moving from to sum on the edge is the line segment in Figure 3.

(1) Find the length of the trapezoid respectively;

(2) Write the coordinates of two points in Figure 3;

(3) Write the functional relationship between and when a point moves on the edge and on the edge respectively (indicate the range of independent variables), and complete the approximate image of the functional relationship in the whole movement in Figure 3.

When the starting point is set to (1) and the point reaches the point and the point just reaches the point, it can be seen from Figure 3 that the area of △ABC at this time is 30. (2) Write the coordinates of two points in combination with the conclusion of (1); (3) Consider two functional relationships about the point above and the point above.

Cultivation of academic ability

1, (Taizhou 07) As shown in the figure, a quadrilateral is a rectangular piece of paper placed in a plane rectangular coordinate system, with points on the axis and points on the axis. Fold the edge so that the point falls on the point of the edge. Folding is known.

(1) and similar? Please explain the reasons;

(2) Find the coordinates of the intersection of the straight line and the axis;

(3) Is there a straight line passing through this point, so that the triangle surrounded by straight line, straight line and axis is similar to the triangle surrounded by straight line, straight line and axis? If it exists, please write its analytical formula directly and draw the corresponding straight line; If it does not exist, please explain why.

2. (Quzhou, Zhejiang Province) The position of the right-angle trapezoidal paper OABC in the plane rectangular coordinate system is shown in the figure. The coordinates of the four vertices are O (0 0,0), A (10/0,0), B (8 8,0) and C (0 0,0) respectively, and the point T is on the line segment OA (not coincident with the end point of the line segment). Fold the paper to make the point stand out.

(1) Find the number of times ∠OAB, and find the functional relationship between S and T when point A' is on line AB;

(2) When the figure of the overlapping part of the paper is quadrilateral, find the value range of t;

(3) Is there a maximum value for S? If it exists, find this maximum value and find the value of t at this time; If it does not exist, please explain why?

3. (Yancheng, Jiangsu) As shown in the figure, in the plane rectangular coordinate system, it is known that △AOB is an equilateral triangle with point A as its vertex.

The coordinate of is (0,4), point B is in the first quadrant, and point P is the moving point on the X axis. Connect AP, rotate △AOP counterclockwise around point A, so that the sides AO and AB overlap, and thus get △ABD.

(1) Find the analytical formula of straight line AB;

(2) When point P moves to point (0), find the length of DP at this time and the coordinates of point D;

(3) Whether there is a point P, so that the areas of △OPD are equal, and if there is, the coordinates of the point P satisfying the conditions are requested; If it does not exist, please explain why.

4. (Leshan, Sichuan) In the plane rectangular coordinate system, the side AB of △ABC is on the X axis, OA >；; OB, a circle with a diameter of AB passes through point C. If the coordinate of C is (0,2) and AB = 5, the abscissas Xa and XB of two points A and B are two equations about X:

(1) Find the values of m and n;

(2) If the straight line where the ∠ACB bisector is located intersects the X axis at point D, find the analytical expression of the linear function corresponding to the straight line;

(3) Let point D be a straight line, and rays Ca and CB (except point C) intersect at point M and point N respectively, so whether the value is a fixed value, if so, find a fixed value, if not, please explain the reason.