Technology: a2=m4+n4-2m2n2
b2=4m2n2
c2=m4+n4+2m2n2
a2+b2=c2
So right angle ~
2. If x+y=3 xy=-5, then y/x+x/y=?
Do it yourself, and the simple "/"is to divide.
First, multiple-choice questions (3 points for each small question, ***24 points)
1. 10 The students' weights are 4 1, 48, 50, 53, 49, 50, 53, 53, 5 1, 67 (unit: kg) respectively.
The poor basis is ()
A.27 B. 26 C. 25 D. 24
2. The number of trees planted in five greening groups in a school is as follows: 10, 10, 12, x, 8. It is known that the mode of this set of data is equal to the average, so the median of this set of data is ().
A.8 B. 9 C. 10 D. 12
3. The height measurement results of 50 students in a class are as follows:
Height1.51.521.531.541.551.571.58/.
Number113434468106
The mode and median height of the students in this class are () respectively.
A. 1.60， 1.56 B. 1.59， 1.58 C. 1.60， 1.58 D. 1.60， 1.60
4. If the variance of a set of data is 2, then the variance of a new set of data 2, 2, 2 is ().
A.2b 4c 8d 16
5. Class A and Class B held a computer Chinese character input competition. The statistical results of the number of Chinese characters input by students per minute are as follows:
Average variance of the median number of class participants
a 55 149 19 1 135
b 55 15 1 1 10 135
After analyzing the above table, a classmate came to the following conclusion:
(1) Class A and Class B have the same average scores; (2) The number of excellent students in Class B is more than that in Class A (excellent students input 150 Chinese characters per minute); (3) The fluctuation of class A's performance is greater than that of class B, and the above conclusion is correct ()
A.⑵⑶b .⑵c .⑶d .⑶⑶。
6. If the average of sample 1, 2, 3, 5 and x is 3, then the variance of the sample is ().
A.3 B. 9 C. 4 D. 2
7. There are two classes in the eighth grade of a school. A math test, there are 52 students in class one, with an average score of 75, and 50 students in class two, with an average score of 76.65. In this exam, the average score of the two classes is ().
A.78.58 B.75.8 1 C
8. A shoe store tried to sell a new type of women's shoes. The sales during the trial sale period are as follows:
Model 22 22.5 23 23.5 24 24.5 25
Quantity/double 3 5 10 15 8 3 2
For the manager of this shoe store, he is most concerned about what kind of shoes sell well, so the following statistics are most meaningful to the manager of the shoe store ()
A. Mean B. Mode C. Median D. Variance
Fill in the blanks (4 points for each small question, 24 points for * * *)
9. In a knowledge contest, the scores of students in Group A and Group B are as follows:
Score 50 60 70 80 90 100
mankind
Count a 251013146
B 4 4 16 2 12 12
Then: =, =
10. In a shooting practice, Party A and Party B shot at the target five times each, and the number of hits was as follows:
The number of target rings for armored shooting is 7 8 6 8 6.
The number of B-ray target rings is 9 5 6 7 8.
Then the shooting performance is relatively stable:
1 1.8 (2) Class randomly investigated the pocket money of 10 students in order to correctly guide students to establish a correct concept of consumption. The statistical chart is as follows:
The pocket money is above 3 yuan (including 3 yuan).
The proportion of students is,
six
four
Average daily allowance for students in this class.
three
About 100 yuan. 2
1
1 2 3 4 5 6 7 8 9 10
12. In order to investigate the traffic volume of a certain road section, the number of vehicles passing through the intersection at the same time every day for 30 days was recorded, including 284 vehicles in 4 days, 290 vehicles in 4 days, 12 vehicles in 3 days, 14 vehicles in 3 days, so this 30.
13. The daily minimum temperature for five consecutive days was measured in Xiaofang, and the following table was obtained after sorting:
Date one two three four five variance average temperature
The lowest temperature is 1 3 2 5.
three
There are two data accidentally contaminated by ink, namely.
14. At the party of two schools in a certain place, A and B were performed by actors of 10, and their ages (unit: years old) were as follows: A program:131415 655.
Procedure B: 5 5 6 6 6 6 7 7 50 52
The model of a is that the age fluctuation of actors is small.
Third, answer question Y (number of people)
15.( 12) Nowadays, the decline of teenagers' vision level has caused
Only with the concern of the whole society can we understand the 3000 students in a school.
Visual acuity, select some students from it.
Sampling survey, draw a straight square with the obtained data.
The figure (the height of the rectangle indicates the number of people in this group) is as shown on the right:
Answer the following questions:
(1) How many students were sampled in this sampling survey?
(2) What are the students' vision patterns?
Within range?
(3) If the visual acuity is 4.9, 5.0, 5. 1 or above, it is normal.
3.95 4.25 4.55 4.85 5.155.45 times (visual acuity)
How many students have normal eyesight in this school?
16.(8 points) In order to estimate the number of fish in the pond, a professional fish farmer first caught 100 fish for marking, and then put them back into the lake. After a period of time, after the marked fish and the fish were completely mixed, he fished again five times. The record is as follows: 90 fish were caught for the first time, of which 1650 was marked. Second fishing 100, of which 9 were marked; Catch 120 fish for the third time, in which 12 fish is marked; The fourth fishing 100, of which 9 were marked; The fifth time, 80 fish were caught, including 8 marked fish. How many fish are there in the pond?
17. (12) In the early morning of August 29th, 2004, in the Olympic women's volleyball final, China made a big reversal after losing two sets first, and finally defeated the Russian women's volleyball team 3-2 to win the championship, which was the second time that the women's volleyball team reached the top of the Olympic Games from 1984. The picture below shows the technology of this key battle.
Statistics: 74
(1) What are the total scores of China and Russian teams respectively?
Less? It is known that the score of the fifth game is15:12. Please calculate.
After China's exit, the average score of the Russian team in the first four games.
(2) Both Chinese and Russian teams scored 23 points.
What are the "most" events? 15
(3) What information can you get from the above picture? (Write 14.
Just give two)
2
18.( 10) A company recruits staff, and interviews and written tests are conducted on two applicants, A and B. The interview content includes physical fitness and eloquence, and the written test content includes professional level and innovation ability. Their scores (100) are as follows:
Candidate interview written test
Innovative ability of professional level of body eloquence
A 86 90 96 92
B 92 88 95 93
(1) If the company thinks that the ratio of physique, eloquence, professional level and innovation ability is 5: 5: 4: 6 according to the nature of business and job requirements, please calculate the average score of Party A and Party B to see who will be admitted.
(2) If the company believes that according to the nature of business and job requirements, interview scores account for 5%, eloquence accounts for 30%, written test scores account for 35%, and innovation ability accounts for 30%, then who do you think the company should admit?
five
four
three
19.( 10) The monthly sales of salespersons is 2 1.
If the amount is X (unit: ten thousand yuan) and X < 15, it meansno..
13 14 15 16 17 18 19 20 2 1 22 23 24 25 28
Competent, 15 ≤ x < 20 basically competent, 20 ≤ x < 25 competent, and x≥25 excellent. (1) Find the percentage of sales staff at four levels and make statistics with a pie chart. (2) The median, mode and average monthly sales of all competent and excellent salespeople.
Reference answer to test questions
1~8
A D B B
9 ~1480,256-50%, 2.8
306 4 and 2 15, a
15.( 1) 150 (2)4.25~4.55 (3) 1400
Article 16. 1000
17.( 1) 1 18, 1 12.25.75,25
(2) Attack score
(3) Omission
18.( 1)90.8,9 1.9; second
(2)92.5,92. 15; first
19.( 1) omitted
(2)22,20 22.3
Review of Grade Eight Mathematics in the Next Term (4)
Class name, student number score
First, multiple-choice questions (3 points for each small question, ***24 points)
1. The correct one in the following propositions is ().
A. A quadrilateral with its diagonal bisected is a diamond B. A quadrilateral with its diagonal bisected and equal is a diamond.
C. quadrilaterals with diagonal lines perpendicular to each other are diamonds.
2. There is an open space of isosceles trapezoid ABCD in the flower field, and the midpoint of each side is E, F, G and H respectively. Diagonal AC= 10 meter. Now if you want to enclose a quadrilateral EFGH field with a fence, the total length of the fence should be ().
A.40m m B.30m m C.20 d.10m
3. In the trapezoid, ABCD, AD‖BC, diagonal AC⊥BD, AC= 10, BD=6, then the area of the trapezoid is ().
A. 30 BC 65438+ 60 AD
4. As shown in the figure, it is known that rectangular ABCD, R and P are on DC and BC respectively.
Point, e and f are the midpoint of AP and RP respectively, when P is in BC.
When r does not move from b to c, the following conclusion holds.
Yes ()
A. the length of line segment Ef increases gradually. B. the length of line segment Ef decreases gradually.
C. the length of the line segment EF remains unchanged. D. the length of line segment EF cannot be determined.
5. Parallelogram, rectangle, square, isosceles trapezoid, right angle
Trapezoidal, non-axisymmetric graphics have ().
A. 1
6. As shown in the figure, the two diagonals in ABCD intersect at point O, and through rotation,
After translation, the triangle * * * that can overlap in the figure has ().
A.2 to B.3 to C.4 to D.5.
7. If the circumference of a diamond is eight times that of it, then a set of adjacent angles is ().
A.30 and 150 B.45 and 135 C.60 and 120 D.80 and 100.
8. In rectangular ABCD, AB = 3 and BC = 4, then the distance from point A to diagonal BD is ().
A.B.2 C. D
Fill in the blanks (3 points for each small question, *** 18 points)
9. In parallelogram ABCD, DB = DC, ∠ C = 70, AE ⊥ BD is in E, then ∠DAE= degrees.
10. As shown in the figure, BD is the diagonal of parallelogram ABCD, and points E and F are on BD, so that quadrilateral AECF can be made.
It is a parallelogram, and one more condition needs to be added. (Fill in only one)
(9 maps) (10 maps)
1 1. As shown in the figure, a parallelogram is divided into four small parallelograms with areas of, and respectively. When a CD slides from left to right along AB in a parallelogram, the relationship with the size is.
12. If the trapezoid has an area of 12c and a height of 3cm, the length of the midline is.
13. The quadrilateral of the diagonal is a diamond.
14. In trapezoidal ABCD, DC‖AB, DC+CB=AB, and ∠ A = 5 1, then the number of ∠B is.
Three. solve problems
15.( 10 minute) Known: As shown in the figure, in the parallelogram ABCD,
E and f are two points on the diagonal AC, AE = cf
Verification: DE=BF E
16.( 18 points) It is known that, as shown in the figure, D is the midpoint of BC side of DE ⊥ AC and DF ⊥ AB △ ABC.
The vertical feet are e and f respectively, and BF=CE.
It is proved that (1)△ABC is an isosceles triangle;
(2) When ∠ A = 90, try to judge that the quadrilateral AFDE is
What kind of quadrilateral, prove your judgment.
17.( 10 minute) As shown in the figure, it is known that the straight line m‖n, A and B are two points on the straight line N, and C and P are two points on the straight line M.
Point. (1) Please write each pair of triangles with equal area in the chart:
.
(2) If A, B and C are three fixed points, then point P moves on m..
So whenever point P moves to any position, there will always be.
Area equal to △ABC;
The reason is:
18.( 10 minute) As shown in the figure, in the diamond ABCD, e is the midpoint of AD.
The extension line from EF⊥AC to CB is at F.
Proof: AB and EF are equally divided.
19.( 14 minutes) As shown in the figure, make three equilateral △ABD, △BCE and △ACF on the same side of BC.
Please answer the following questions:
(1) Verification: the quadrilateral ADEF is a parallelogram;
(2) When △ABC satisfies any condition, the quadrilateral ADEF is a rectangle.
Reference answer to test questions
1~8 DC AC
British Broadcasting Corporation
9 ~ 14 20 be = df (not unique) = 1
4 Divided by 78 degrees perpendicular to each other.
15. Omit
16.( 1) omitted
②AFDE is square.
17.( 1) △ ABC and △ABP, △AOC and △BOP, △CPA and △ CPB;
(2)△ ambulatory blood pressure,
(3) Same bottom and equal height
18. Omit
19.( 1) omitted
(2) 150
Selected exercises
1. True or false
In a triangle, if the center line of one side is equal to half of this side, then the angle subtended by this side is a right angle.
(2) Proposition: "In a triangle, there is an angle of 30? Then the opposite side is half of the other side. " The counter-proposition holds.
(3) The converse theorem of Pythagorean Theorem is: If the sum of squares of two right-angled sides is equal to the square of the hypotenuse, then this triangle is a right-angled triangle.
⑷ Delta ABC is a right triangle when the ratio of three sides is 1: 1:.
Answer: right, wrong, wrong, right;
2. The opposite sides of ∠A, ∠B and ∠C in 2.△ ABC are A, B and C respectively, and the false proposition in the following proposition is ().
A. if ∠ c-∠ b = ∠ a, then △ABC is a right triangle.
B if C2 = B2-A2, then △ABC is a right triangle and ∠ c = 90.
C if (c+a) (c-a) = B2, then △ABC is a right triangle.
D if ∠ a: ∠ b: ∠ c = 5: 2: 3, then △ABC is a right triangle.
Answer: d
3. The following four line segments can't form a right triangle is ()
A.a=8，b= 15，c= 17
B.a=9，b= 12，c= 15
C.a=，b=，c=
D.a:b:c=2:3:4
Answer: d
4. It is known that in △ABC, the opposite sides of ∠A, ∠B and ∠C are A, B and C respectively, which are the following lengths. Do you judge whether a triangle is a right triangle? And point out which angle is the right angle?
⑴a=，b=，c =； ⑵a=5，b=7，c = 9；
⑶a=2，b=，c =； ⑷a=5，b=，c= 1。
Answer: (1) Yes, ∠ b; (2) no; (3) Yes, ∠ c; 4 yes, ∠ a.
5. State the inverse proposition of the latter proposition and judge whether the inverse proposition is correct.
(1) if a3 > 0, then a2 > 0；;
(2) If the angle of a triangle is less than 90, the triangle is an acute triangle;
(3) If two triangles are congruent, their corresponding angles are equal;
(4) Two line segments that are symmetrical about a straight line must be equal.
Answer: (1) If A2 > 0, A3 > 0；; False proposition.
(2) If the triangle is an acute triangle, then one angle is acute; True proposition.
(3) If the corresponding angles of two triangles are equal, then the two triangles are congruent; False proposition.
(4) Two equal line segments must be symmetrical about a straight line; False proposition.
6. Fill in the blanks.
Every proposition exists, but not every theorem exists.
The inverse theorem of "two straight lines are parallel and the internal dislocation angles are equal" is.
(3) In △ABC, if A2 = B2-C2, then △ABC is a triangle and a right angle; If A2 < B2-C2, ∠B is.
(4) If a = m2-N2, b=2mn and c = m2+N2 in △ABC, then △ABC is a triangle.
Answer: (1) inverse proposition, inverse theorem; ⑵ Internal dislocation angles are equal and two straight lines are parallel; (3) right angle, ∠B, obtuse angle; (4) right angle.
Xiao Qiang walked 80 meters to the east on the playground, then 60 meters, and then 100 meters back to his original place. Xiao Qiang walked 80 meters to the east on the playground, and then walked 60 meters.
Answer: due south or due north.
7. If three sides of a triangle are (1) 1, 2; ⑵ ; ⑶32,42,52 ⑷9,40,4 1;
⑸(m+n)2- 1，2(m+n)，(m+n)2+ 1； What constitutes a right triangle is ()
A.2 B.3 C.4 D.5
Answer: b
8. If the three sides A, B and C of △ABC satisfy (A-B) (A2+B2-C2) = 0, then △ABC is ().
A. isosceles triangle;
B. right triangle;
C. isosceles triangle or right triangle;
D. isosceles right triangle
Answer: c
9. As shown in the picture, there is a 2-meter-long shadow rod CD standing on the playground. Its shadow length BD is 4m in the morning and its shadow length AD is 1 m at noon. Can A, B and C form a right triangle? Why?
Answer: Yes, because BC2=BD2+CD2=20, AC2=AD2+CD2=5, AB2=25, BC2+AC2= AB2.
10. As shown in the picture, a ship of unknown nationality entered the offshore of China. Two patrol boats A and B of our navy immediately intercepted them from bases A and B, which were 13 nautical miles apart, and arrived at place C at the same time six minutes later to intercept them. It is known that patrol boat A sails at a speed of 120 nautical miles per hour, while patrol boat B sails at a speed of 50 nautical miles per hour with a heading of 40 northwest. Q
Answer: From △ABC is a right triangle, we can know that ∠ cab+∠ CBA = 90, so ∠ cab = 40, and the course is 50 northeast.
1 1. As shown in the picture, Xiaoming's father opened a quadrangular plot near the fish pond and planted some vegetables. Dad asked Xiaoming to calculate the area of the land so as to calculate the output. Xiao Ming found a roll of rice ruler and measured AB = 4m, BC = 3m, CD = 13m and DA = 12m. .
Tip: Connect AC. AC2 = AB2+BC2 = 25, AC2+AD2=CD2, so ∠CAB=90? ,
S quadrilateral =S△ADC+S△ABC=36 square meters.
12. Known in △ABC, ∠ ACB = 90, CD⊥AB in D, CD2=AD? BD。 Prove that Delta △ABC is a right triangle.
Prompt: ∫ac2 = ad2+Cd2, BC2=CD2+BD2, ∴AC2+BC2=AD2+2CD2+BD2=
AD2+2AD？ BD+BD2=(AD+BD)2=AB2,∴∠ACB=90。
13. In △ABC, AB= 13cm, AC= 24cm, BD= 5cm ... It is proved that △ABC is an isosceles triangle.
Tip: Because AD2+BD2=AB2, AD⊥BD is judged by the vertical line in the line segment, and AB = BC.
14. Known: as shown in the figure, ∠ 1=∠2, AD=AE, D is a point above BC, BD=DC, AC2 = AE2+Ce2. Verification: AB2 = AE2+Ce2.
Hint: ∠ E = 90 and AC2 = AE2+CE2；; From △ ADC △ AEC, AD=AE, CD=CE, ∠ ADC = ∠ BE = 90. Judging from the median vertical line, AB=AC, then AB2 = AE2+Ce2.
15. Given that three sides of △ABC are A, B, C, a+b=4, ab= 1, c=, try to determine the shape of △ABC.
Tip: The right triangle is proved by algebraic method, because (a+b)2= 16, a2+2ab+b2= 16, ab= 1, so A2+B2 = 14. And c2= 14.
In March 2009, the eighth grade mathematics monthly examination paper of Hezhuang Middle School.
Class name exam number score
First, multiple-choice questions, (3 points for each small question, ***30 points) Please choose carefully, and you will definitely choose the right one!
1 The simplest common denominator of the fraction is ()
a、x+ 1 B、x- 1 C 、( x+ 1)D、x(x
2, the common denominator of fractional numerator is ()
a、x B、x C、3x D、 12x
3. The solution of the fractional equation+=-is ()
A, x= 1 B, x=- 1 c, no solution d, x =
4. If the fractional equation +=2 has no solution, the value of m is ().
a、 1 B、2 C、3 D、4
5. Calculate the result of the score γ. Yes ().
a、2x B、C、D
6. The result of expressing 0.00000207 by scientific counting method is ().
a、2.07× 10 B、2.07× 10 C、207× 10 D、2.07× 10
7. When the speed of the ship in still water is 30 km/h, the time taken to sail downstream along the river 100 km is equal to the time taken to sail upstream for 60 km. If the speed of the river is x km/h, the equation listed is ().
a、B、C、D、
8 when k > 0 and y < 0, the image of inverse proportional function y= is in ().
A, the first quadrant b, the second quadrant c, the third quadrant d and the fourth quadrant
9. Among the following functions, the one where y is the inverse proportional function of x is ().
a、y=- B、y= - C、y= D、y=
10, for y= the following statement is wrong ()
A, the points that the image must pass through (1, 2 2) b and y decrease with the increase of x c, the image is in the first and third quadrant d, if X > 1, Y < 2.
Fill in the blanks: (3 points for each small question, ***24 points) Think carefully and fill in carefully, and you will succeed!
1 1, if the score is meaningful, then x _ _ 12, if the score, then x = _ _ _
13, without changing the value of the fraction and turning all the symbols of m into positive numbers, then _ _ _
14. There are three points (x, y), (x, y) and (x, y) on the image with inverse proportional function Y=.
X < 0 < x < x, then the relationship between y, y and y is _ _ _ _ _.
15, simplify the score to _ _ _ _ _.
16, the radius of a bacterium is 4× 10 meter, expressed in decimal as _ _ _ _ _ meter.
17, if x+, then x = _ _ _ _
18, it is known that one branch of the image of function y= is in the fourth quadrant, so the value range of k is _ _ _ _ _ _ _ _ _ _ _ _.
Third, calculation: (6 points for each small question, ***20 points) Be careful, or you will make mistakes!
19、 20、
2 1 、( 22 、( x- 1-
Fourth, solve the equation: (6 points for each small question, *** 10) I believe you can solve it well, but pay attention to the steps!
23、 24、
5. Solving application problems by listing equations: (10) You must be careful and you can do it well!
25. Students from Class 8 (1) and Class 8 (2) of Hezhuang Middle School participated in afforestation. It is known that 8 classes (1) plant 5 more trees every day than 8 classes (2), and the time required for 8 classes (1) to plant 80 trees is equal to the time required for 8 classes (2) to plant 70 trees.
Six, (each small question 10, ***20) This problem is not difficult, you should seriously consider it, you can certainly do it perfectly!
26. A branch of an image with a known inverse scaling function y= is in the fourth quadrant.
(1), which quadrant is the other branch of the image in? What is the range of the constant k?
(2) Take points A(a, b) and B(a) on a branch of this function image. If A > A, what is the size relationship between B and B?
(3) If points C(m, n) and D(m) are both on this function image, and m < 0, m > 0, what is the size relationship between n and n?
27. Summer is coming. Taihe Servant Commercial Building is going to install a batch of air conditioners. If 60 air conditioners are installed every day, it will take 20 days to install them.
(1) If X units are installed every day, the number of days required is Y. Write the functional relationship between Y and X. ..
(2) According to the formula, if 80 air conditioners are installed every day, how many days will it take?
(3) Due to the sudden hot weather, it takes 12 days to pack. How many units should be installed at least every day?
Tips: You must check the test paper carefully, but don't send it in a hurry, or you will regret it! Develop the habit of caution! Exercise 2
First, fill in the blanks:
1. Turn the following score into the simplest score: (1) = _ _ _ _ _ _; (2) =_______; (3) =________.
2. The basic nature of the score is: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _;
Represented by letters: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
3. If a=, the value of is equal to _ _ _ _. 4. Calculation = _ _ _ _ _ _.
5. then what? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Multiple choice question:
6. Don't change the value of the fraction, so that all the coefficients of the fraction can become integers, and the numerator and denominator should be multiplied by ().
10
7. The following equation: ① =? ; ② = ; ③ = ? ;
④ = ? , established is ()
A.①② B.③④ C.①③ D.②④
8. Do not change the value of the fraction, so that the coefficient of the highest term of the numerator denominator is positive, and the correct one is ().
A.B. C. D。
9. Fraction,,, is the simplest fraction ()
1。
10. According to the basic properties of the score, the score can be converted into ()
A. BC? D.
1 1. Among the following statements, the correct one is ().
A.= ; b . =； c . =； D. =
12. Among the following categories, the correct one is ().
A.B. = 0 C. D。
13. The simplest common denominator in the formula is ()
A.(x？ 1)2 B.(x？ 1) 3 C.(x？ 1) D.(x？ 1)2( 1? x)3
Answer the question:
14. Convert the following scores into scores with the same denominator:
( 1) , , ; (2) , , .
15. About integral:
( 1) ; (2) .
16. Comprehensive score:
( 1) , ; (2) , .
17. It is known that a2 -4a+9b2+6b+5=0. What? The value.
18. Given x2+3x+ 1=0, find the value of x2+.
19. Given x+ =3, the value of.
Sorry, geometry can't be uploaded, only so much can be loaded. Please forgive me!
I've seen it, and it's very good, especially there are several proof questions behind it.