A/b=b/(a-b) on the math proportion problem in grade three.
a^2-ab-b^2=0
a/b=( 1+√5)/2
I don't know where you are a student. But on the whole, the methods are similar. First, find out the simulation questions of each district. Seriously sum up the methods. Let's say a few first. 1, if there is no right angle, generally extend the radius to form a diameter to obtain a right angle, and then chamfer. 2. If there is a right angle, prove that the angle is a right angle by four methods: parallel isosceles triangle with three lines in one, the median line of one side is equal to half of this side, and chamfering. These are basically the ideas.
Examples and exercises in the math circle of grade three, as long as you answer the questions ~ ~ ~ ~
2007 Junior High School Mathematics Competition Examination Questions (Shandong Division) 2008 National Junior High School Mathematics Competition Shandong Division
Preliminary examination questions of junior high school mathematics competition in Shandong Province in 2007
1. Multiple-choice questions (8 questions in this question, 6 points for each question, out of 48 points): Only one of the options given in the following questions is correct. Please fill in the code of the correct option in the brackets after the question.
1. It is known that the function y = x2+1–x, and the point P(x, y) is on the image of this function. Then, the point P(x, y) should be on the () of the rectangular coordinate plane.
(a) first quadrant (b) second quadrant (c) third quadrant (d) fourth quadrant
2. There are m red balls, 10 white balls and n black balls in a box, and each ball is the same except the color. If you choose a ball from them, the probability of getting a white ball is the same as the probability of not getting a white ball, then the relationship between m and n is ().
(A)m+n = 10(B)m+n = 5(C)m = n = 10(D)m = 2，n = 3
3. Our province stipulates that a junior high school math contest will be held on the last Sunday of June every year 1 1, and the date of next year's junior high school math contest is ().
(a)165438+1October 26th (b)165438+/October 27th (c)165438+1October 29th.
4. There are two points A (–2,2), B (3 3,2), C and C in the plane rectangular coordinate system. If △ABC is a right triangle, then the point C that meets the conditions is ().
1 (B)2 (C)4 (D)6。
5. As shown in the figure, there are points E and F on the sides BC and CA of the regular triangle ABC, which satisfy the following requirements.
BE = CF = a，EC = FA = b(a & gt； b)。 When BF divides AE equally, the value of ab is ().
(A)5– 12(B)5–22(C)5+ 12(D)5+22
6. A company ordered 22 lunches in a fast food restaurant, which cost 140 yuan. There are three kinds of lunches: A, B and C, and the unit prices are 8 yuan, 5 yuan and 3 yuan respectively. Then the possible different sorting schemes are ().
1 (B)2 (C)3 (D)4。
7. As we all know, a > 0, b>0 and a (a+4b) = 3b (a+2b). Then the value of a+6ab–8b2a–3ab+2b is ().
(A) 1(B)2(C) 19 1 1(D)2
8. As shown in the figure, in trapezoidal ABCD, ∠ d = 90, and m is the midpoint of AB, if
CM = 6.5, BC+CD+DA = 17, then the area of trapezoidal ABCD is ().
20 (B)30 (C)40 (D)50
Fill in the blanks (4 small questions in this question, 8 points for each small question, out of 32 points): Answer the question.
Fill in directly on the horizontal line of the corresponding topic.
9. As shown in the figure, in the rhombic ABCD, ∠ A = 100, m and n are AB and BC respectively.
If MP⊥CD is at p, then the degree of ∠NPC is.
10. if the real number a satisfies a3+a2–3a+2 = 3a–1a2–1a3,
Then a+ 1A =
1 1. As shown in the figure, in △ABC, ∠ BAC = 45, AD⊥BC in D, if BD = 3, CD.
= 2, then S⊿ABC =
12. The linear function y =–33x+1intersects the X axis and the Y axis respectively.
Points A and B form a square ABCD (e.g.
Figure). There is a point P(a, 12) in the second quadrant, which satisfies the square ABCD of s △ ABP = S.
Then a =
Third, answer the question (this question ***3 small questions, 20 points for each small question, out of 60 points)
13, as shown in the figure, points Al, Bl, C 1 are respectively on the sides of AB, BC and CA of △ABC.
And aa1ab = bb1BC = cc1ca = k (k
The perimeter of is p 1. Verification: p 1
14. There are several students living in a dormitory of a school, one of whom is the head of the dormitory. On New Year's Day, each student in the dormitory gave each other a card, and each dormitory administrator also gave the person in charge of the dormitory a card, so * * * used 5 1 card. Ask how many students live in this dormitory.
15. if a 1, a2, …, an are all positive integers, and A 1
Reference answer:
I baddc cbb ii . 9.50 10.2 or–3 1 1. 15 12.32–8。
Three. 13. Omit 14. 6 students 15. Omit.
Mathematics (Proportion) Solution of Grade Three: According to the meaning of the question,
b+c=ak
a+c=bk
a+b=ck
Triple addition
2(a+b+c)=k(a+b+c)
A+b+c=0 or k=2.
When a+b+c=0, k=(b+c)/a=-a/a=- 1.
So k=2 or k=- 1.
Guangdong junior high school math review free website can go to the middle school subject network, where there are many free resources to download.
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Mathematics review knowledge framework (knowledge points, examples) 1, related concepts of circle;
(1), determine that the elements of a circle are the center and radius.
(2) A line segment connecting any two points on a circle is called a chord. The chord passing through the center of the circle is called the diameter. The part between any two points on a circle is called an arc. An arc smaller than half a circle is called a bad arc. An arc larger than half a circle is called an optimal arc. In the same circle or equal circle, arcs that can overlap each other are called equal arcs. The vertex is on the circle, and the angle at which both sides intersect the circle is called the circumferential angle. You can draw a circle through the three vertices of a triangle, and only one can be drawn. The circle passing through the three vertices of a triangle is called the circumscribed circle of the triangle, and the center of the circumscribed circle of the triangle is called the circumscribed circle of the triangle, and the center of the circumscribed circle is the intersection of the vertical lines of each side of the triangle. The radius of the circumscribed circle of a right triangle is equal to half of the hypotenuse. The circle tangent to each side of the triangle is called the inscribed circle of the triangle, the center of the inscribed circle of the triangle is called the inner circle of the triangle, and this triangle is called the circumscribed triangle. The inner circle of the triangle is the intersection of bisectors of three inner angles of the triangle. The radius of the inscribed circle of a right triangle satisfies:
2. Related properties of circles
Theorem (1) In the same circle or equal circle, if the central angles are equal, then the arcs it faces are equal, the chords it faces are equal, and the chords it faces are equally spaced. It is inferred that in the same circle or equal circle, if one group of quantities of two central angles, two arcs, two chords or the distance between two chords are equal, then the other groups of quantities of their pairs are equal respectively.
(2) Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord. Inference1(i) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects the two arcs opposite to the chord. (Ⅱ) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord. (iii) bisect the diameter of the arc subtended by the chord, bisect the chord vertically and bisect the other arc subtended by the chord. Inference 2 The arcs sandwiched between two parallel chords of a circle are equal.
(3) Theorem of circumferential angle: the circumferential angle of an arc is equal to half the central angle of the arc. Inference 1 In the same circle or equal circle, the circumferential angles of the same arc or equal arc are equal, and so are the arcs with equal circumferential angles. Inference 2 The circumferential angles of semicircles or diameters are all equal, all equal to 90. The chord subtended by a circumferential angle of 90 is the diameter of the circle. Inference 3 If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.
(4) Determination and properties of the tangent: Determination theorem: The straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle. Property theorem: the tangent of a circle is perpendicular to the radius passing through the tangent point; A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point; A straight line perpendicular to the tangent through the tangent point must pass through the center of the circle.
(5) Theorem: Three points that are not on the same straight line determine a circle.
(6) The length of the line segment between a point on the tangent of a circle and the tangent point is called the tangent length from the point to the circle; Tangent length theorem: two tangents of a circle can be drawn from a point outside the circle, and their tangents are equal in length. The connecting line between this point and the center of the circle bisects the included angle between the two tangents. (7) The quadrangles inscribed in the circle are diagonally complementary, and one outer angle is equal to the inner diagonal; The sum of the opposite sides of the circumscribed quadrangle is equal;
(8) Chord angle theorem: the chord angle is equal to the circumferential angle of the arc pair it clamps.
(9) Proportional line segments related to a circle: the theorem of intersecting chords: the product of two intersecting chords in a circle is equal to the length of two lines divided by the intersection. If the chord intersects the diameter vertically, then half of the chord is the proportional average of two line segments formed by its separate diameters. Secant theorem: the tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the middle term in the length ratio of the two lines where this point intersects the secant. Draw two secants of a circle from a point outside the circle, and the product of the lengths of the two lines from that point to the intersection of each secant and the circle is equal.
(10) Two circles are tangent, and the connecting line intersects the tangent point; Two circles intersect, and the connecting line bisects the common chord vertically.
Math problem 1. Math problem: There is a lamp by the river in the park. At point B 3 meters above the water surface, the elevation angle of the lamp is 30, and the depression angle of the image of the lamp in the river is measured.
2. Xiao Wang, a salesman of a dairy processing factory, sent 10 boxes of milk powder to the supermarket, with 20 bags in each box and 400g in each bag. When he was about to return to the factory, he suddenly received a phone call from the factory department, saying that one of the 10 cases of milk powder was out of order because of the canning machine, and each bag was short of 20 grams, so he was asked to take the missing box back for exchange immediately, but the supermarket was busy at that time.
3. Mathematical problems
Find a pattern
72.90.2430.8 10.90.().()
4. Find common math problems about circles.
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5. In △ABC, it is known that A = 60, and the maximum and minimum edge is the equation X? Two real roots of -9X+8=0, find the side length BC?
6. A proposition is related to a positive integer n. If n=k, the proposition holds. If it can be inferred that n=k+ 1, the proposition also holds. Now it is known that when n=5, the proposition does not hold, so the proposition is
A n=6 holds, B n=6 holds, C n=4 holds, and D n=4 holds.
Let the radius of the inscribed circle be R.
So s cycle =πr?
The side length of a regular triangle is 2 root numbers 3 * R.
Height is 3r
∴S△=2 root number 3*r*3r/2=3 root number 3r?
∴P (area from needle to inscribed circle) =πr? /3 3r? = π radical 3/9
Huanggang famous volume Qingdao edition is good, Huanggang Qingdao edition is formulated according to local educati