(2) When the line segment moves, the area of the quadrilateral is, and the movement time is. Find the functional relationship between the area of quadrilateral and the moving time, and write the range of independent variables.
2. As shown in the figure, in the trapezoid, the moving point starts from the point and moves along the line segment to the end point at a speed of 2 unit lengths per second; At the same time, the moving point starts from the point and moves along the line segment to the end point at the speed of 1 unit length per second. Set the moving time to seconds.
(1).
(2) The value of when.
(3) Try to explore why it is an isosceles triangle.
3. As shown in the figure, in the plane rectangular coordinate system, the quadrilateral OABC is trapezoidal, OA∨BC, the coordinate of point A is (6,0), the coordinate of point B is (4,3), and point C is on the positive semi-axis of the Y axis. The moving point m moves on OA and is sent from point O to point A; Moving point n moves on AB, from point A to point B, two moving points start at the same time, and the speed is 1 unit length per second. When one of the points reaches the end point, the other point stops immediately, and the movement time of the two points is t (seconds).
(1) Find the length of line segment AB; What is the value of t, MN∑OC?
(2) Let the area of △CMN be s, and find the resolution function between s and t,
And point out the range of independent variable t; Is there a minimum value for s?
If there is a minimum, what is it?
(3) After AC, is there such a t, where MN and AC are perpendicular to each other?
If it exists, find out the t value at this time; If it does not exist, please explain why.
2. (Hebei Volume) As shown in the figure, in Rt△ABC, ∠ c = 90, AC = 12, BC = 16, the moving point P moves from point A to point C along the AC edge at a speed of 3 units per second, and the moving point Q moves from point C to point B along the CB edge at a speed of 4 units per second.
(1) Let the area of quadrangle PCQD be y, and find the functional relationship between y and t;
(2) What is the value of t, and the quadrilateral PQBA is a trapezoid?
(3) Is there a time t for PD∑AB? If it exists, find the value of t; If it does not exist, please explain the reason;
(4) through observation, painting or origami, guess whether there is time t to make PD⊥AB? If it exists, please estimate the time period of the value of t in brackets (0 ≤ t ≤1; 1 < t≤2； 2 < t≤3； 3 < t≤4)； If it does not exist, please briefly explain why.
3. (Jining, Shandong) As shown in the figure, A and B are points on the positive semi-axis of the X axis and the Y axis respectively. The lengths of OA and OB are 2 of the equation x2- 14x+48 = 0 (OA > OB) respectively. The straight line BC bisects ∠ABO, the X axis is at point C, and P is the moving point on BC. Point P moves from point B to BC at a speed of 1 unit per second.
(1) Let the areas of △APB and △OPB be S 1 and S2 respectively, and find the value of S 1∶S2;
(2) Find the analytical formula of BC line;
(3) let pa-po = m, and the moving time of point p is t.
(1) when 0 < t ≤, try to find the range of m;
② When t >, what do you think is the value range of m (only the conclusion is required)?
4. In China, there are two moving points, P and Q, starting from point A and point B respectively, in which point P moves along AC to end C at the speed of 1cm/s; Point Q moves along BC to terminal C at the speed of 1.25cm/s/s, and the passing point P is PE∨BC, and the passing point AD is connected with EQ. Let the moving time of the fixed point be x seconds.
The lengths of (1)AE and DE are expressed by an algebraic expression containing X;
(2) When point Q moves on BD (excluding points B and D), let the area be, find the functional relationship with the month, and write the range of independent variables;
(3) When it is a value, it is a right triangle.
5. In (Hangzhou) right-angled trapezoid, the height is (as shown in figure 1). The moving point starts from the point at the same time, and the point moves and stops from point to point, and the speed of the two points is the same. And when the point reaches the point, the point just reaches the point. Let's assume that the time from the point at the same time is and the area is (as shown in Figure 2). Establish rectangular coordinate system with abscissa and ordinate respectively. When a point moves from to sum on the edge, the function image of sum 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.
6. (Jinhua) As shown in figure 1, in the plane rectangular coordinate system, the known point is on the positive semi-axis, and the moving point moves from point to point on the line segment at the speed of unit per second, assuming that the moving time is seconds. Take two points on the axis as equal sides.
(1) Find the analytical formula of the straight line;
(2) Find the length of the equilateral (expressed by algebraic expression), and find the value when the equilateral vertex moves to coincide with the origin;
(3) If we take the midpoint as the edge and make a rectangle as shown in Figure 2 in it, let the area of the overlapping part of the equilateral and rectangle be the point on the line segment, and find the functional relationship of the sum when the second is 0, and find the maximum value.
7. As shown in figure 1, put two identical right triangles ABC and DEF, with points C and F coincident and BC and d F on a straight line, where AC=DF=4 and BC = EF = 3. Keep Rt△ABC still and let Rt△DEF translate to the left along CB until point F and point B coincide. Let FC=x, two triangles.
(1) As shown in Figure 2, when x=, what is the value of y?
(2) As shown in Figure 3, when the point E moves to AB, find the values of X and Y;
(3) Find out the functional relationship between Y and X;
8. (Chongqing curriculum reform volume) as shown in figure 1, a triangular piece of paper ABC, ∠ ACB = 90, AC = 8, BC = 6. Cut this paper into two triangles along the center line CD of the hypotenuse AB (as shown in Figure 2). Translate the paper along the straight line (AB) (the points are always on the same straight line).
(1) When panning to the position shown in Figure 3, guess the quantitative relationship between sum in the figure and prove your guess;
(2) If the translation distance is and the overlapping area is, please write the functional relationship between sum and the range of independent variables;
(3) Whether there is such a value for the conclusion in (2); So the area of the overlapping part is equal to the original area? If it does not exist, please explain why.
1. In trapezoidal ABCD, AD∨BC, ∠ B = 90, AD=24cm, AB=8cm, BC=26cm, and the moving point P starts from point A and moves along the edge of AD to point D at the speed of 1 cm/s; The moving point Q starts from point C and moves to point B at a speed of 3cm/s along the edge of CB.
It is known that P and Q start from A and C at the same time, and when one of them reaches the end point, the other one stops moving. Assuming that the movement time is t seconds, ask:
What is the value of (1)t, and the quadrilateral PQCD is a parallelogram?
(2) Could the quadrilateral PQCD be a diamond at a certain moment? Why?
(3) At what value is t, the quadrilateral PQCD is a right trapezoid?
(4) At what value of t, the quadrilateral PQCD is an isosceles trapezoid?
2. As shown in the right figure, in the rectangular ABCD, AB=20cm, BC=4cm, dot.
P starts from A and moves along the dotted line A-B-C-D at a speed of 4cm/s, and point Q starts from C.
Start moving along the edge of CD at the speed of1cm/s. If points P and Q come from A and C respectively,
At first, when one of the points reaches point D, the other point also stops moving, assuming it is moving.
When the time is t(s), what is the value of t, and the quadrilateral APQD is also a rectangle?
3. As shown in the figure, in the isosceles trapezoid, ∩, AB= 12 cm, CD=6cm, the point moves along the edge at a speed of 3cm per second from the beginning, and moves along the edge of CD at a speed of 1cm per second from the beginning. If point P and point Q start from point A and point C at the same time, the movement stops when one of them reaches the finish line. Let the exercise time be t seconds.
(1) Verification: When t=, the quadrilateral is a parallelogram;
(2) 2) Is it possible for PQ to divide diagonal BD equally? If so, find out when t is the value, and PQ divides BD equally; If not, please explain the reasons;
(3) If △DPQ is an isosceles triangle with PQ as the waist, find the value of t. ..
4. As shown in the figure, in △ABC, point O is the moving point on the side of AC, the intersection point O is the straight line MN//BC, the bisector of the intersection point of MN is at point E, and the bisector of the intersection point of outer corners is at F. ..
(1) Make way:
(2) When the point O moves to where, the quadrilateral AECF is a rectangle? And prove your conclusion.
(3) If there is a little o on the AC side, let the quadrilateral AECF be a square, AEBC=62, and find the size of.
5. As shown in the figure, in the rectangle ABCD, AB=8, BC=4, fold the rectangle along AC, and the point D falls on the point D', and find the area of the overlapping part ⊿AFC.
6. As shown in the figure, there are four moving points P, Q, E and F, which start from the four vertices of the square ABCD and move to points B, C, D and A at the same speed along AB, BC, CD and DA.
(1) Try to judge that the quadrilateral PQEF is a square and prove it.
(2) 2) Whether PE always crosses a certain point, and explain the reasons.
(3) When the vertex of the quadrilateral PQEF is located,
What is its smallest and largest area? How much is each?
7. It is known that in trapezoidal ABCD, AD∨BC, AB = DC, diagonal AC and BD intersect at point O, E is the moving point on the side of BC (point E does not coincide with points B and C), EF∨BD intersects at point F, and EG∨AC intersects at point G. 。
(1) Verification: the perimeter of quadrilateral EFOG is equal to 2ob；; ;
⑵ Please change the condition of the above topic "AD∨BC, AB = DC in trapezoidal ABCD" to another quadrilateral, and other conditions remain unchanged, so that the conclusion that the perimeter of the quadrilateral EFOG is equal to 2 OB still holds, and draw the figure of the adapted topic, and write what is known, verified and needs no proof.
As shown in the figure, in the right-angled trapezoidal ABCD, ABC, ∠ ABC = 90, known AD = AB = 3, BC = 4, and the moving point P starts from point B and moves to point C at a uniform speed along BC line; The moving point Q starts from the point D and moves to the point A at a constant speed along the straight line DA. The light passing through the point Q and perpendicular to the AD intersects with the point BC at the points N, P and Q, all at the speed of 1 unit length per second. When point Q moves to point A, point P and point Q stop moving at the same time. The time for the set point q to move is t seconds.
(1) Find the length of NC and MC (expressed by algebraic expression of t);
(2) When t is what value, does the quadrangle PCDQ form a parallelogram?
(3) Is there a moment when the ray QN just bisects the area and perimeter of △ABC at the same time? If it exists, find the value of t at this time; If it does not exist, please explain the reason;
(4) Inquiry: When t is a value, why is △PMC an isosceles triangle?
9. (Shandong Qingdao Curriculum Reform Volume) As shown in Figure ①, there are two right-angled triangles ABC and EFG(A with the same shape (point A coincides with point E). It is known that AC = 8 cm, BC = 6 cm, ∠ C = 90, EG = 4 cm, ∠ EGF = 90, and O is the midpoint on the hypotenuse of △EFG.
As shown in Figure ②, if the whole △EFG starts from the position in Figure ① and moves in the direction of ray AB at the speed of 1cm/s, and △EFG moves, then point P starts from the vertex G of △EFG and moves to point F on the right-angled side GF at the speed of 1cm/s, and when point P reaches point F, it stops moving.
(1) What is the value of x, OP∑AC?
(2) Find the functional relationship between Y and X, and determine the range of independent variable X. 。
(3) Is there a moment of 13∶24 for the ratio of quadrilateral OAHP area to △ABC area? If it exists, find the value of x; If it does not exist, explain why.
(Reference data:1142 =12996,1152 =13225,1162 =/.
Or 4.42 = 19.36, 4.52 = 20.25, 4.62 = 2 1. 16).
10, known: As shown in the figure, △ABC is an equilateral triangle with a side length of 3cm, the moving point.
P and Q start from point A and point B at the same time, and move at a constant speed along AB and BC respectively.
Move, their speed is 1cm/s, when point P reaches point B, both P and Q are.
The point stops moving. Let the movement time of point P be t(s) and answer the following questions:
(1) When t is what value, is △PBQ a right triangle?
(2) Let the area of quadrilateral APQC be y(cm2) and find the sum of y and t.
Relationship; Is there a moment t that makes the area of quadrilateral APQC two-thirds of the area of △ABC? If it exists, find the corresponding t value; If it does not exist, explain the reasons;