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Mathematical problems that need to be solved in the process
As shown in the figure, in the trapezoidal ABCD, DC∨AB, ∠ A = 90, AD=6 cm, DC=4 cm, the slope I of BC = 3: 4, the moving point P starts from A and moves along the AB direction to point B at a speed of 2 cm/s, and the moving point Q starts from point B and moves along the B? c? Direction d moves to point d, and both moving points start at the same time. When one of the moving points reaches the end point, the other moving point also stops. Let the moving time of the fixed point be t seconds.

(1) Find the side length BC;

(2) When t is what value, PC and BQ are equally divided;

(3) Connect PQ, let the area of △PBQ be y, explore the functional relationship between y and t, and find out what is the value of t and the maximum value of y? What is the maximum value?

As shown in the figure, AM∨BN, ∠ AE=x ∠ BC=y. 90, AB=4, point D is a moving point on the ray AM (point D does not coincide with point A), point E is a moving point on the line segment AB (point E does not coincide with points A and B), connecting de, and the intersection point E is the vertical line of de.

(1) When AD= 1, find the functional relationship between y and x, and write its definition domain;

(2) Under the condition of (1), take the midpoint f of the line segment DC and connect EF. If EF=2.5, find the length of AE;

(3) If the moving points D and E always meet the condition AD+DE=AB when moving, please explore: Does the perimeter of BCE change with the movement of points D and E? Please explain the reason.

As shown in the figure, in the rectangular coordinate system, the bottom AB of trapezoidal ABCD is on the X axis, and the endpoint D of the bottom CD is on the Y axis. The expression of straight line CB is y = -4/3x+ 16/3, and the coordinates of point A and point D are (-4,0) and (0,4) respectively. The moving point P starts from point A and runs on AB at a constant speed. The speed is 1 unit per second. When one of the moving points reaches the end point, they stop moving at the same time. When the point p moves t (seconds), the area of △OPQ is S (except for the moving point that cannot form △OPQ).

(1) Find the coordinates of point B and point C;

(2) Find out the functional relationship between S and T;

(3) When the value of t is what, what is the maximum value of s? And find the maximum value.

As shown in the figure, in the right-angled trapezoidal ABCD, AD∨BC, ∠ A = 90, AB= 12, BC=2 1, and AD = 16. The moving point P starts from point B and moves in the direction of ray BC at a speed of 2 units per second, and moves point Q..

(1) Let the area of △DPQ be s, and find the functional relationship between s and t;

(2) When t is what value, is the quadrilateral PCDQ a parallelogram?

(3) When t is what value, ①PD=PQ, ② DQ = PQ.

As shown in the figure, the side length of the square ABCD is 8cm, the moving point P starts from point A and moves from point A to point B at a constant speed of 1cm/ sec (point P does not coincide with points A and B), and the moving point Q starts from point B and moves along the dotted line BC-CD at a constant speed of 2cm/ sec. Point P and point Q start at the same time, and when point P stops, point Q also stops. Connect AQ.

(1) When point Q moves on BC line, how long after point P leaves, ∠BEP=∠BEQ?

(2) Let the area of △APE be ycm2 and AP=xcm, find the resolution function of Y about X, and write the definition domain of the function.

(3) When 4 < x < 8, find the range of function value y 。

As shown in the figure, in the rectangle ABCD, AB=3 and BC=4, the moving point P starts from point D and moves along DA to endpoint A, while the moving point Q starts from point A and moves along diagonal AC to endpoint C. When the intersection point P is PE∨DC and AC intersects with point E, the moving speed of moving points P and Q is 1 unit length per second and the moving time is x seconds.

(1) Find the functional relationship between y and x;

(2) Inquiry: When x is what value, is the quadrilateral PQBE a trapezoid?

(3) Are there such points P and Q, and the triangle whose vertices are P, Q and E is an isosceles triangle? If it exists, request all values of x that meet the requirements; If it does not exist, please explain why.

As shown in the figure, Rt△ABC, ∠ A = 90, AB= 10, AC=5. If the moving point P starts from point B and moves to point A along the BA line, the moving amount is 2 unit lengths per second. If the intersection point P is PM∨BC and the intersection point AC is at point M, let the moving time of the moving point P be x seconds and the length be AM.

(1) Find the functional relationship between Y and X, and write the range of the independent variable X;

(2) When x is what value, the area s of △BPM has the maximum value. What is the maximum value?

The side length of a square ABCD is 8 cm. The moving point P starts from point A and moves from point A to point B along the side AB at a constant speed of 1cm/sec (point P does not coincide with points A and B). The moving point q starts from the point b and moves at a constant speed of 2cm/sec along the dotted line BC-CD. Point p and point q start at the same time. When point P stops moving, point Q also stops moving. Connect AQ and pay BD.

(1) When point Q moves on BC line, after how long point P leaves, ∠BEP and ∠BEQ are equal;

(2) When the Q point moves on the BC line, it is proved that the area of △BQE is twice that of △APE;

(3) Let the area of △APE be y, try to find the resolution function of y about x, and write the definition domain of the function.

I hope it helps you.