Look for the eighth grade mathematics final test question 1 in the second volume on Baidu

18. It is known that the side BC of the rectangle ABCD is on the X-axis, and E is the diagonal BD The coordinates of the midpoint, points B and D are respectively

B (1, 0), D (3, 3). The image of the inverse proportional function y= passes through point A,

(1) Write the coordinates of point A and point E;

(2) Find the analytical formula of the inverse proportional function;

(3) Determine whether point E is on the graph of this function

19. It is known that: CD is the height on the hypotenuse of , and , , , (as shown in the figure). Verification:

Reference answers

1. D 2. B 3. A 4. D 5. C 6. B 7. C 8. C 9. C 10. B

11.3

12. -1 or y=-x-1 or y=

13.1

14.19.1cm, 164.3 cm

15.1

16. 2x-1, 3

17. Solution: (1) The number of people in the contaminated place is 11. Suppose the number of donations in the contaminated area is yuan, then

11 1460=50×38

Solution=40

Answer: (1) The contaminated area The number of people in the contaminated area is 11, and the donation amount for the contaminated area is 40 yuan.

(2) The median donation amount is 40 yuan, and the mode of donation amount is 50 yuan.

18. Solution: (1) A (1, 3), E (2, 32)

(2) Suppose the required functional relationship is y=kx

p>

Substituting x=1 and y=3, we get: k=3×1=3

∴ y=3x is the required analytical formula

(3 ) When x=2, y=32

∴ Point E (2, 32) is on the graph of this function.

19. Proof: The left side

∵ is in a right triangle,

and ∵ is the right side

∴

That is to say, it is proved that:

The final test question 3 of the eighth-grade mathematics volume 2 of the People's Education Press

1. Multiple-choice questions

1. The fifth national census The results show that the total population of our country has reached 1,300,000,000 people. Using scientific notation to express this number, the correct result is ( )

A. 1.3×108 B． 1.3×109 C. 0.13×1010 D. 13×109

2. Without changing the value of the fraction, average the coefficients in the fraction into integers. The result is ( )

A, B, C, D,

3. If the voltage across a resistor of a certain value is 5 , and the current passing through it is 1 , then the approximate image of the current passing through this resistor changing with the voltage across it is (hint: ) ( )

4. If x and y in the fraction are expanded by 2 times, then the value of the fraction ( )

A. Expand by 4 times; B. Expand by 2 times; C. No change; D reduced by 2 times

5. As shown in the figure, there is a right-angled triangle piece of paper with two right-angled sides. Now fold the right-angled edge along a straight line so that it falls on the hypotenuse and coincides with .

Then it is equal to ( )

, , , ,

6. The vertices A, B, C, and D in the rectangle ABCD are arranged in the clockwise direction. If they are in the plane Cartesian coordinate system, The coordinates corresponding to the two points B and D are (2, 0), (0, 0) respectively, and the two points A and C are symmetrical about the x-axis. Then the coordinates corresponding to the point C are

(A) (1, 1 (B) (1, -1) (C) (1, -2) (D) (2, -2)

7. The following figures are centrally symmetrical figures, but The figure that is not axially symmetrical is (　).

(A) Square (B) Rectangle (C) Rhomboid (D) Parallelogram

8. As shown in the figure, E, F, G , H are the midpoints of the four sides of quadrilateral ABCD respectively. To make quadrilateral EFGH a rectangle, the conditions that quadrilateral ABCD should meet are (　).

(A) One set of opposite sides is parallel and the other set of opposite sides Not parallel (B) Diagonals are equal

(C) Diagonals are perpendicular to each other (D) Diagonals bisect each other

9. Which of the following statements is wrong ( )

A. The diagonals of a parallelogram are equal. B. The diagonals of an isosceles trapezoid are equal.

C. A parallelogram with two equal diagonals is a rectangle

D. A quadrilateral with perpendicular diagonals is a rhombus

10. If the graph of the function y=2 x +k intersects the positive half-axis of the y-axis, then the quadrant of the graph of the function y= is ( )

A. The first and second quadrants B. The third and fourth quadrants C. The second and fourth quadrants D. The first and third quadrants

11. If it represents an integer , then the integer a can have ( )

A. 2 C. 3 D. 12. As shown in the figure, the square cardboard The side length is 4, and points E and F are the midpoints of AB and BC respectively. If it is cut along the dotted line in the left picture and assembled into a "small villa" as shown in the right picture below, then the area of ??the shaded part in the picture is ( )

A, 2 B, 4 C, 8 D, 10

2. Fill in the blanks

13. The graph of the known proportional function and the graph of the inverse proportional function If the abscissa of an intersection point is p>

Machine Tool A: =10, =0.02; Machine Tool B: =10, =0.06, it can be seen that: ________ (fill in A or B) the performance of the machine tool is good.

15. There is a tree 9 A meter-high tree with a 1-meter-tall child under the tree. If the tree breaks (not broken) 4 meters above the ground, the child will be safe at least 1 meter away from the tree.

16. Write an inverse proportional function so that its function value in the quadrant increases with the increase of the independent variable. The analytical formula of this function can be . (Just write one)

17. The picture is a trademark pattern designed by Sunshine Company for a certain product. The shaded part in the picture is red. If the area of ??each small rectangle is 1, then the red part The area is 5.

18. As shown in the figure, in □ABCD, AE and CF are the angle bisectors of ∠BAD and ∠BCD respectively. According to the existing graphics, please add a condition to make the quadrilateral AECF a rhombus, then add A condition of can be (just write one, no other "points" and "lines" can be added to the picture).

19. Known: In the isosceles trapezoid ABCD, AD ∥BC, diagonal AC⊥BD, AD=3cm, BC=7cm, then the height of the trapezoid is _______cm

20. As shown in the figure, the lengths of the diagonals of rhombus ABCD are 2 and 5 respectively. , P is any point on the diagonal AC (point P does not coincide with points A and C), and PE∥BC intersects AB at E, PF∥CD intersects AD at F, then the area of ??the shaded part is _______.

3. Answers and proofs

21. ⑴ Calculation:

⑵ Simplification:

22. The known function y=y1 y2, Among them, y1 is directly proportional to x, y2 is inversely proportional to x-2, and when x=1, y=-1; when x=3, y=5, find the analytical formula of this function.

23. First simplify , and then ask you to take a set of values ??and substitute them into the evaluation.

24. Solving equations

25. As shown in the figure, in the square ABCD, E is a point on the side of CD, F is a point on the extension line of BC, CE=CF, ∠FDC= 30°, find the degree of ∠BEF.

26. As shown in the figure, the Meteorological Observatory in City A measured that the center of the typhoon was at B, 320km due west of City A, moving in the BF direction of 60° north-east at a speed of 40km per hour. The area within 200km is the area affected by typhoons.

⑴ Is City A affected by this typhoon? Why?

⑵ If City A is affected by this typhoon, how long will it take for City A to be affected by this typhoon?

27. As shown in the figure, the image of the linear function y=kx b intersects the image of the inverse proportional function y= ax at two points A and B, intersects the x-axis at point C, and intersects the y-axis at point C. D, it is known that OA=5, the coordinates of point B are (12, m), draw the AH⊥x axis through point A, the vertical foot is H, AH= 12 HO

(1) Find the inverse proportional function and the analytical formula of a linear function;

(2) Find the area of ??△AOB.

28. As shown in the figure, in the quadrilateral ABCD, AC=6, BD=8 and AC⊥BD sequentially connect the midpoints of each side of the quadrilateral ABCD to obtain the quadrilateral A1B1C1D1; then connect the midpoints of each side of the quadrilateral A1B1C1D1 in sequence. point, we get the quadrilateral A2B2C2D2...and continue in this way to get the quadrilateral AnBnCnDn.

(1) Prove that the quadrilateral A1B1C1D1 is a rectangle;

(2) Write the areas of the quadrilateral A1B1C1D1 and the quadrilateral A2B2C2D2 ;

(3) Write the area of ??the quadrilateral AnBnCnDn;

(4) Find the perimeter of the quadrilateral A5B5C5D5.

Reference answers

1. Multiple choice questions

1. B2, B 3, D 4, B 5, B 6, B 7, D8, C9, D 10, D 11, D 12, B 13, (-1 , 2) 14. A 15, 4 16, y=-1x (the answer is not unique) 17, 518, AE=AF (the answer is not unique) 19, 125 20, 2.5

21. Solution: ⑴ Original formula = 4-8×0.125 1 1 =4-1 2 =5 ⑵-m-2

22. Solution: Suppose

; ∵ at that time, ; at that time, ,

23. Solution: Original formula

Evaluation: Take a set of values ??and substitute them into the evaluation.

24. Solution:

Multiply both sides of the equation at the same time:

Solution: Test: At that time,

is the original fraction Solution of Eq.

25, 105° The proband △BCE≌△DCF yields ∠EBC=∠FDC=30°, and ∠BEC=60°, so it can be obtained.

26. Solution: ⑴ It will be affected by typhoon, because the distance from P to BF is 160kmlt; 200km;

⑵The impact time is 6 hours.

27. Solution:

The ∵ point on the graph of the inverse proportional function

is

,

∴The analytical formula of a linear function is

28 (1) Prove that ∵points A1 and D1 are the midpoints of AB and AD respectively, and ∴A1D1 is the median line of △ABD

∴A1D1 ∥BD, , the same principle: B1C1∥BD,

∴ ∥, =, ∴The quadrilateral is a parallelogram

∵AC⊥BD, AC∥A1B1, BD∥, ∴A1B1⊥ That is, ∠B1A1D1=90°

∴The quadrilateral is a rectangle

(2) The area of ??the quadrilateral is 12; the area of ??the quadrilateral is 6;

(3) Quadrilateral The area of is 4x, and the width is 3x, then

The solution is; ∴ ;

∴Perimeter of the rectangle = .

Method 2: Area of ??the rectangle/area of ??the rectangle Area

= (Perimeter of the rectangle) 2/(Perimeter of the rectangle) 2

That is: 12 = (Perimeter of the rectangle) 2: 142

∴Perimeter of a rectangle =

Eighth grade math final test question 4 in volume 2

1. Fill in carefully and make the final decision (among the four options given in each question, Only one is correct, please choose the correct option and fill in the correct option in the answer sheet)

1. Students all know that the hives built by bees are both strong and material-saving. So do you know the thickness of the hive comb? In fact, the honeycomb thickness of the hive is only about 0.000073m. This data is expressed in scientific notation as ( )

A, B, C, D,

2. If the two diagonals of a quadrilateral are equal, the quadrilateral is called Diagonal quadrilateral.

Which of the following shapes is not a diagonal quadrilateral ( )

A. Parallelogram B. Rectangle C. Square D. Isosceles trapezoid

3. The highest temperature in a certain place for 10 consecutive days The statistics are as follows:

Maximum temperature (℃) 22 23 24 25

Number of days 1 2 3 4

The median and mode of this set of data are respectively ( )

A, 24, 25 B, 24.5, 25 C, 25, 24 D, 23.5, 24

4. Among the following operations, the correct one is ( )

A, B, C, D,

5. Among the following sets of numbers, the triangle with a, b, c as sides is not Rt△ ( )

A. a=2, b=3, c=4 B. a=5, b=12, c=13

C. a=6, b=8, c=10 D. a= 3. b=4, c=5

6. The range of a set of data 0, -1, 5, x, 3, -2 is 8, then the value of x is ( )

A, 6 B, 7 C, 6 or -3 D, 7 or -3

7. The known point (3, -1) is a point on the hyperbola, then each of the following The point that is not on the hyperbola is ( )

A, B, C, (-1,3) D, (3,1)

8. Which of the following statements is correct? ( )

A. The mode, median and mean of a set of data cannot be the same number

B. The mean of a set of data cannot be the same as the mean of this set of data. Any number in the data is equal

C. The median of a set of data may not be equal to any number of this set of data

D. Mode, median and mean Numbers describe the fluctuation size of a set of data from different angles

9. As shown in Figure (1), the length of the diagonal of the rectangle is known to be. Connect the midpoints, , , of each side to obtain a quadrilateral, then The perimeter of the quadrilateral is ( )

A, B, C, D,

10. If the equation about x has no solution, then the value of m is ( )

A, -3 B, -2 C, -1 D, 3

11. In square ABCD, diagonal AC=BD=12cm, point P is any point on side AB One point, then the sum of the distances from point P to AC and BD is ( )

A, 6cm B, 7cm C, cm D, cm

12, as shown in Figure (2) shows that the area of ??the rectangle ABCD is 10, its two diagonals intersect at the point, and draw a parallelogram with AB and as the adjacent sides. The diagonals of the parallelogram intersect at the point. Similarly, draw a parallelogram with AB and as the adjacent sides. ,..., and so on, then the area of ??the parallelogram is ( )

A, 1 B, 2 C, D,

2. Fill in the details carefully, I believe you can fill them in Fast and accurate

13. If the image of the inverse proportional function

In each quadrant, y decreases as x increases, then the value of k can be _______ (just write a qualified k value)

14. Eighth grade in a middle school Two classes A and B with equal numbers participated in the same mathematics test. The average score and variance of the two classes are points, points, , respectively. Then the one with more uniform scores is ________ (fill in "Class A" or "Class B").

15. As shown in Figure (3), in □ABCD, E and F are points on sides AD and BC respectively. If a condition_____________ is added, the quadrilateral EBFD is a parallelogram.

16. As shown in Figure (4), it is a line statistical chart of a set of data. The average of this set of data is , and the range is .

17. As shown in Figure (5), there is a right-angled trapezoidal part ABCD, AD∥BC, oblique waist DC=10cm, ∠D=120°, then the length of the other waist AB of the part is _ ______cm;

18. As shown in Figure (6), the quadrilateral is a rhombus with a perimeter of , and the coordinates of the point are, then the coordinates of the point are.

19. As shown in Figure (7), if two pieces of isosceles right triangle paper of the same size are used to make a jigsaw puzzle, the following graphics will be obtained: ① Parallelogram (excluding rectangle, rhombus, square); ② Rectangle (excluding square); ③ Square; ④ Equilateral triangle; ⑤ Isosceles right triangle, among which the shapes that can be assembled are __________ (just fill in the serial number).

20. Any positive integer n can be decomposed like this: (s, t are positive integers, and s≤t), if the absolute value of the difference between the two factors in all such decompositions of n The smallest one is called the best decomposition and is specified. For example: 18 can be decomposed into 1×18, 2×9, 3×6, this is . Combining the above information, give the following statements: ①; ②; ③; ④ If n is a perfect square number, then the correct statement is _________. (Just fill in the serial number)

3. Start Use your brain, you will definitely get it right (the answer should include written explanation, proof process or deduction steps)

21. Solve equations

22. Simplify first, then evaluate, where x=2.

23. 50 students from Class 1 of Grade 8 of a certain school participated in the 2007 Jining Mathematics Quality Monitoring Examination. The performance statistics of the whole class are as follows:

Score (points) 71 74 78 80 82 83 85 86 88 90 91 92 94

Number of people 1 2 3 5 4 5 3 7 8 4 3 3 2

Please answer the following based on the information provided in the table Questions:

(1) What are the mode and median of the test scores of the students in this class?

(2) Zhang Hua’s score in this exam is 83 points. Can we say that Zhang Hua’s score is above the average of the class? Try to explain the reason.

24. As shown in Figure (8), 5 small squares of the same size are arranged into the shape as shown in the figure. Now move one of the small squares. Please click on the

Picture Draw figures that meet the following requirements in (8-1), Figure (8-2), and Figure (8-3) respectively. (Indicated by shading)

(1) Make the resulting figures axially symmetrical graphics, rather than centrally symmetrical graphics;

(2) Make the resulting graphics a centrally symmetrical graphics, rather than an axially symmetrical graphics;

(3) Make the resulting graphics both an axially symmetrical graphics, It is a centrally symmetrical figure again.

25. A youth research institution randomly investigated the amount of pocket money of 100 students in a certain school during the winter vacation (the amount was rounded up to an integer) in order to conduct research, analysis and guide students to establish correct consumption. Observe. The frequency distribution table shown in the figure below is now made based on the survey data.

(1) Please complete the frequency distribution table and frequency distribution histogram;

(2) ) The study believes that students who spend more than 150 yuan should be given suggestions on thrift and reasonable consumption. How many students out of the 1,200 students in the school should be given this suggestion?

(3) What other information can you get from the following charts? (Write at least one)

Group (yuan) Group median (yuan) Frequency Frequency

0.5～50.5 25.5 0.1

50.5～100.5 75.5 20 0.2

100.5～150.5

150.5～200.5 175.5 30 0.3

200.5～250.5 225.5 10 0.1

250.5～300.5 275.5 5 0.05

Total 100

26. As shown in the figure, the image of the linear function and the image of the inverse proportional function intersect at two points M and N.

(1) Find the analytical formulas of the inverse proportional function and the linear function according to the conditions in the figure;

(2) When x is what value, the value of the linear function is greater than the value of the inverse proportional function?

27. As shown in the figure, fold one side AD of the rectangle ABCD so that point D falls at point F on the BC side. It is known that AB=8cm and BC=10cm. Find the length of CE?

28. As shown in the figure, in the trapezoid ABCD, AD∥BC, ∠B=90°, AD=24 cm, BC=26 cm, the moving point P starts from the point A Starting from point D, it moves in the AD direction at a speed of 1cm/s. The moving point Q starts from point C and moves in the CB direction towards point B at a speed of 3cm/s. Points P and Q start from point A and point C at the same time respectively. When one point reaches the endpoint, the other point stops moving.

(1) How long has it taken for the quadrilateral PQCD to become a parallelogram?

(2) How long has passed before the quadrilateral PQBA became a rectangle?

(3) How long has it taken for the quadrilateral PQCD to become an isosceles trapezoid?

Reference answers

1. Multiple choice questions (3 points × 12 = 36 points)

Question number 1 2 3 4 5 6 7 8 9 10 11 12

Answer B A A D A C D C A B A D

2. Fill in the blanks (3 points × 8 = 24 points)

13. kgt ; Any value of 4 (the answer is not unique); 14. ___Class A___; 15. The answer is not unique; 16. 46.5, 31;

17. cm; 18. (0, 3) ; 19. __①③⑤__; 20. __①③④__.

3. Use your brain, you will definitely get it right (***60 points)

21. (6 points) Solution: both sides of the equation Multiply together to get:

Solve to get:

Check: Substitute =0

So -2 is the increased root of the original equation, and the original equation has no solution.

22. (6 points) Solution: Original formula =

Put x=2 into the original formula = 8

23. (8 points) (1) Plural The number is 88, and the median is 86;

(2) No, the reason is omitted.

24, (6 points)

25, (9 points )

(1) Briefly

(2) (Name)

(3) Briefly

26. (8 points) Solution : (1) The analytical formula of the inverse proportional function is:

The analytical formula of the linear function is:

(2) When or , the value of the linear function is greater than the value of the inverse proportional function.

27. (8 points) CE=3

28. (9 points) (1) (3 points) Suppose that the quadrilateral PQCD is a parallelogram, that is, PD=CQ,

So we get

(2) (3 points) Assume that the quadrilateral PQBA is a rectangle, that is, AP=BQ, so we get

(3) (3 points) Suppose After that, the quadrilateral PQCD is an isosceles trapezoid. (The process is omitted)

Final exam of mathematics in the second semester of the second grade of junior high school

(Time: 90 minutes; full score: 120 points)

1. Multiple choice questions: (3 points × 6 = 18 points)

1. As shown in the figure, the mass of each weight in the right pan of the balance is 1g, then the mass of object A is m ( The value range of g) can be expressed as ( ) on the number axis

2. The following figure is a schematic diagram of the principle of small hole imaging. According to the dimensions marked in the figure, the candle formed in the dark box The length of a CD is ( )

A. 1/6cm B. 1/3cm C. 1/2cm D. 1cm

3. Which of the following propositions is true ( )

A. If x, then -2x 3lt; -2y 3

B. Two straight lines are intercepted by a third straight line, and the angles are equal

D. Congruent figures must be similar figures, but similar figures are not necessarily congruent figures

5. The picture below is a frequency distribution histogram of the number of heartbeats per minute in a physical examination of a class of students in the second grade of junior high school. (The times are all integers).

It is known that there are only five students in the class whose heartbeats are 75 beats per minute. Please observe the picture below and indicate which of the following statements is wrong ( )

A. The data 75 falls in group 2

B. The frequency of group 4 is 0.1

D. The data 75 must be the median

6. Two people A and B start from place A at the same time and ride bicycles to In place B, it is known that the distance between place AB is 30 kilometers. A walks 3 kilometers more per hour than B and arrives 40 minutes before B. Suppose B walks x kilometers per hour, the equation can be written as ( )

2. Fill in the blanks: (3 points × 6 = 18 points)

7. Factoring: x3 -16x=____________.

8. As shown in the figure, it is known that AB//CD, ∠B=68o, ∠CFD=71o, then ∠FDC=________ degree.

9. An equal number of students from Class A and Class B took the same math test. The average score and variance of the class are as follows:

10. Point P is the hypotenuse of Rt△ABC At a point on AB that is different from A and B, draw a straight line PE through point P to cut △ABC, so that the cut triangle is similar to △ABC. Please draw a straight line that meets the conditions in the figure below, and briefly explain it below the corresponding figure. The vertical or parallel positional relationship between the straight line PE and the side of △ABC.

Positional relationship: ____________ __________ __________

12. In △ABC, AB=10.

3. Drawing questions: (5 points)

13. Use compasses and rulers to draw, do not write down the method, but keep the traces of drawing.

When Xiao Ming makes a class corner for his class, he must enlarge the graphics on the original picture so that the ratio of the new graphics to the corresponding line segments of the original graphics is 2:1. Please help Xiao Ming complete this task.

4. Answer the question: (***79 points)

14. (7 points) Please simplify first, and then choose a formula that makes the original formula meaningful, and you Substitute your favorite numbers into the evaluation:

15. (8 points) Solve the following set of inequalities, represent the solution set on the number axis, and write its integer solution.

16. (8 points) Xishui Food Factory produces a kind of fructose at a cost of 24 yuan per kilogram. There are two sales plans:

Option 1: If it is given directly to the owner If the factory is located in a sales department in this city and sells it, the selling price per kilogram will be 32 yuan, but the sales department must pay relevant fees of 2,400 yuan per month;

Option 2: If it is sold directly to local supermarkets, The ex-factory price is 28 yuan per kilogram.

If only one plan can be sold every month, and each plan can sell out the current month's products on a monthly basis, let the factory's monthly sales volume be x kilograms.

(1) If you are the factory director, how should you choose a sales plan to make the factory gain greater profits that month?

(2) When the factory director listened to the summary of each department, the sales director said that he adopted the best plan for sales every month, so he achieved good work performance, but the factory director saw that the accountant sent After receiving the report on the relationship between sales volume and profit for the first quarter (as shown in the table below), you find that the sales volume written in the table is inconsistent with the actual profit delivered. Please find out the discrepancy and calculate the actual sales for the first quarter. total amount.

17. (8 points) Haohao’s mother bought several bottles of yogurt at Yunli Supermarket for 12.50 yuan, but she found at Liqun Supermarket that the same yogurt was 0.2 cheaper per bottle than at Yunli Supermarket. Yuan, so when I bought yogurt the next day, I went to Liqun Supermarket and ended up spending 18.40 yuan. The number of bottles I bought was many times the number of bottles I bought the first time. I asked her about the capacity of the store for the first time. How many bottles of yogurt did you buy at the supermarket?

18. (8 points) The ideological and moral construction of minors has attracted more and more attention from society.

A youth research institute randomly investigated the amount of pocket money spent by 100 students in a school in Dalian during the winter vacation (the amount was rounded up to an integer) in order to guide students to establish a correct concept of consumption. A frequency distribution table and frequency distribution histogram were made based on 100 survey data:

(1) Complete the frequency distribution table and frequency distribution histogram; in the table, A=______, B=______, C= ______

(2) The sample in this question is ____________________________________________.

(3) The institute believes that students who spend more than 150 yuan should be given advice on thrift and thrift. How many of the 1,000 students in the school should be given this advice?

19. (8 points) (1) A student wanted to use the tree shadow to measure the height of the tree. At some point, he measured that the height of the upright pole was 1 meter and the length of the shadow was 0.9 meters, but he went When measuring the tree shadow, he found that the upper part of the tree shadow fell on the wall CD. (As shown in the picture) He measured BC=2.7 meters and CD=1.2 meters. Can you help him find the height of the tree?

(2) Within 24 hours a day, can you help him find other ways to measure (optional include rulers, benchmarks, and mirrors)? Please draw a diagram and combine it with your graphic description:

Experimental equipment used: ____________________________

Line segment whose length needs to be measured: ____________________________

20. (8 points ) A community raised 1,600 yuan and planned to spray paint on a trapezoidal open space with an upper and lower base of 10 meters and 20 meters respectively. As shown in the picture, (1) the unit price of the paint they sprayed on the △AMD and △BMC zones is 8 yuan/m2. When the △AMD zone is full (the shaded part in the picture), it costs 160 yuan. Please calculate the amount of paint sprayed on it. Complete the required fees for the △BMC zone. (2) If the rest of the area is sprayed with two brands of paint, Nili and Yide, with unit prices of 12 yuan/m2 and 10 yuan/m2 respectively, which paint should be chosen to just use up the funds raised?

21. (12 points) Exploration and Innovation:

As shown in the figure: It is known that there are two parallel straight lines AB and CD in the plane, and P is the straight line AB and CD in the same plane Move a little bit outside. (1) When point P moves between AB and CD, and to the left of line segment AC, as shown in Figure (1), what is the relationship between ∠P, ∠A, and ∠C?

Please prove your conclusion:

(2) When point P moves between AB and CD, to the right of the two points of line segment AC, as shown in Figure (2), this What is the relationship between ∠P, ∠A, and ∠C? (No need to prove.) Answer:

(3) As point P moves, can you find two other different types of positional relationships, draw the corresponding graphics, and write ∠ at this time What is the relationship between P, ∠A, and ∠C? Choose one of these to prove.

Practice and application:

Fold a rectangular piece of paper ABCD (as shown in the picture) along EF so that point B falls at B1 within the rectangle and point C falls at C1. B1C1 and DC intersect at point G. Fill in the blanks based on the conclusions of the above exploration:

22. (12 points) Using geometric figures to factorize, the combination of numbers and shapes can help us understand the problem very well.

(1) For example: Add appropriate numbers to the following horizontal lines to make them completely flat.

As shown in the picture above, "x2 8x" is based on the square with side length x, plus two small rectangles with length x and width 4. In order to make it completely flat (that is, the figure becomes a square), a small square with a side length of 4 must be added. That is x2 8x 42=(x 4)2.

Please draw a picture on the horizontal line in the picture below and explain in words how x2-4x _______=(x-______)2 and fill in the blanks.

Explanation:

(2) It is known that the sum of the areas of a square with side length x and a rectangle with length x and width 8 is 9. Look at the picture to find the side length x: (Add the corresponding numbers or algebraic expressions to the letters A, B, C, and x)

A=__________, B=__________

C=__________, x=__________

(3) The perfect square formula can be expressed by the area of ??a plane geometric figure. In fact, some algebraic formulas can also be factored in this form, for example: using area to factorize: a2 4ab 3b2,

So: a2 4ab 3b2=(a b)(a 3b).

Combine this question with the knowledge you have learned about factoring, write an algebraic expression containing the letters a and b, draw geometric figures, and use the geometric figures to write the result of factoring. Provide the following three shapes: a square with side lengths a and b, and a rectangle with length a and width b (use each at least once).