In the isosceles triangle ABCD, the included angle between AB‖CD and diagonal lines AC, BD ∠ AOB is 60, and P, Q and R are the midpoint of OA, BC and OD respectively.
It is proved that △PQR is an isosceles triangle.
A factory wants to build a cuboid water storage tank with a volume of and a depth of. If the cost of each tank bottom is USD and the cost of each tank wall is USD, how to design the tank to minimize the total cost? What is the lowest total cost?
The distance between a and b is several kilometers, so the car travels at a constant speed from a to b, and the speed cannot exceed kilometers per hour. It is known that the transportation cost of an automobile per hour (in yuan) consists of a variable part and a fixed part: the variable part is directly proportional to the square of the speed (kilometers per hour), the proportional coefficient is yuan, and the fixed part is yuan. (1) The whole transportation cost (yuan) is expressed as speed (yuan). (2) At what speed should the car travel to minimize the whole transportation cost?
A 1-meter-long fence encloses a rectangular vegetable garden, with one side against the wall. What is the maximum area of the vegetable garden when the length and width of this rectangle are respectively?
It is known that the sum of two right angles of a right triangle is equal to 10. What is the maximum area of the hypotenuse?
A unit builds a small rectangular house with a building area of12m2 and a back wall. The front building cost 1.200 yuan /m2, the side building cost 800 yuan /m2 and the roof cost 5800 yuan. If the wall height is 3m, and the back cost of the house is not included, how to design the house to minimize the total cost, and what is the minimum total cost?
, a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ is called factorization.
2. The axioms and inferences we have learned to judge two congruent triangles are abbreviated as _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
3. Keep 0.002078 as _ _ _ _ _ _ _ _ _ _ _ _ _ _.
4. Calculate 0.13+(110) 0-10-3 = _ _ _ _ _ _ _ _ _ _.
5. One external angle of the triangle is equal to 1 10, one internal angle is 40, and the other two internal angles of the triangle are _ _ _ _ _ _ _ _ _ _ _.
6. (a-b) n = _ _ _ _ _ (b-a) n (n is an odd number).
7. If one side of a triangle is 9 and the other side is 4, then the value range of the third side is _ _ _ _ _ _ _. If the length of the third side is an integer, it may be _ _ _ _ _ _ _ _ _ _ _.
8. Every term of the polynomial 2πr+2πR contains a common factor * * _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
9. As shown in the figure, AB=AC, AD=AE, ∠ BAC = ∠ DAE: ∠ Abd = _ _ _ _ _.
Sorry, the picture is missing. Hope to understand! )
10. Given that the rational numbers x, y and z satisfy (x2-xy+y2)2+(z+3)2=0, then x3+y3+z3 = _ _ _ _ _ _ _ _ _ _.
Second, multiple-choice questions (3 points for each question, ***30 points)
1, the following factorizations are ()
a、x2-y3 B、a2+b2 C、mx-ny D 、-x2+y2
2, according to the definition, the angle bisector, median line and height line of a triangle is ()
A, straight line B, line segment C and ray D, all of which are wrong.
3.9× 108- 109 is equal to ()
a、 108 B、 10- 1 C 、- 108 D 、- 1
4. If the intersection of three heights of a triangle happens to be a vertex of the triangle, then the triangle is ().
A, acute triangle B, obtuse triangle C, right triangle D, uncertainty
5. Turn 0.0 169a4b6 into a single square, which is ().
a、 1.3a2b3 B、0. 1.3 a2 B2 C、0. 1.3 a2 B3 D、0. 1.3 a2 B4
6. As shown in the figure, ∠ A+∠ B+∠ C+∠ D+∠ E+∠ F is equal to ().
a、480 B、360 C、240 D、 180
Sorry, the picture is missing. Hope to understand! )
7. If (m+n)(m-n)2-mn(m+n)=(m+n)N, then n is ().
a、m2+n2 B、m2-mn+n2 C、m3+mn+n2 D、m2-3mn+n2
8, the following statement is correct ()
A, every proposition has an inverse proposition B, and every theorem has an inverse theorem
C, the inverse proposition of true proposition is true proposition D, and the inverse proposition of false proposition is false proposition.
9. If A, B and C are three sides of a triangle, then the value of the algebraic formula a2-2ab-c2+b2 ().
A, greater than 0 B, equal to 0 C, less than 0 D, uncertain
10, among the following theorems, the inverse theorem is ()
All right angles are equal. B, the vertex angles are equal. C, congruent triangles's corresponding angles are equal.
D, the point on the bisector of the angle is equal to the distance on both sides of the angle.
Three. Factorization: (24 points)
( 1)x4y-xy4(2)ab(C2+D2)+CD(a2+B2)(3) 10 x2-23xy+ 12 y2
(4)(x2+2x)2-7(x2+2x)-8(5)4x 6-3 1x 3-8(6)x4+4
4. Known line segments A and C (A
└————————————┘a
└———————————————————┘b
5. It is known that △ABC, AD is its angular bisector, BD = CD, DE and DF are perpendicular to AB and AC respectively, and the vertical feet are E and F ... Verification: EB = FC (5 points).
Sorry, the picture is missing. Hope to understand! )
6. It is known that a, b and c are positive rational numbers, 3a3+6a2b-3a2c-6abc=0. Verification: a=c(4 points).
Seven, known: AB=AE, ∠B=∠E, BC=ED. Point f is the midpoint of CD. Verification: AF⊥CD(5 points)
Sorry, the picture is missing. Hope to understand! )
Eight, verification; The distance between the two endpoints of one side of a triangle and the center line of this side is equal. (5 points)
Nine, it is known: a2+2a+b2-4b+5=0, how to find the value of ab? (3 points)