On the 1:5000 scale drawing of a rectangular playground, the length of the rectangle is 5cm and the width is 4cm. How many meters is the actual perimeter of the rectangular playground

On the 1:5000 scale drawing of a rectangular playground, the length of the rectangle is 5cm and the width is 4cm. How many meters is the actual perimeter of the rectangular playground

Length: 0.05 * 5000 = 250m
Width: 0.04 * 5000 = 200m
The perimeter is (250 + 200) * 2 = 900m

Mathematicians' stories (in 20-50 words)

Hua Luogeng is a mathematician and academician of Chinese Academy of Sciences. He is mainly engaged in the research and teaching of analytic number theory, matrix geometry, typical group, automorphic function theory, multi complex function theory, partial differential equation, high-dimensional numerical integration and other fields, and has made outstanding achievements

The addition and subtraction of fractions: the square of A-B part of a + the square of B-A part of B

First, B2 / B-A is equivalent to negative B2 / a-b
So the original formula is A2 / A-B minus B2 / a-b
Reduced to (A2-B2) / a-b
（a+b）（a-b）/a-b
The original formula is equal to a + B

Must the height of an equilateral triangle be equally divided into two triangles of equal shape and size? What triangles are the separated figures? (proof)

Yes, according to three lines in one (height, angle, bisector, midline)
Two congruent right triangles

The nature of the two figures obtained by translation is that the two figures before and after translation are------（

Congruent graph
Attached:
Translation is to make all the points on a graph in the same direction in the plane
The movement of distance is called the translational movement of a graph. It does not change the shape and size of the graph. It is isometric isomorphism and is a kind of affine transformation in affine space. It can be regarded as the result of adding the same vector to each point or moving the center of the coordinate system. That is to say, if a known vector is a point in space, it can be translated
After translation, the corresponding line segments are parallel (or collinear) and equal, the corresponding angles are equal, and the line segments connected by the corresponding points are parallel and equal; the translation transformation does not change the shape, size and direction of the figure (the two figures before and after translation are congruent). (1) the shape and size of the figure before and after translation do not change, but the position changes; (2) after the figure translation, the shape and size of the figure are congruent, The line segments connected by corresponding points are parallel and equal (or on the same line) (3) multiple translations are equivalent to one translation. (4) the figure after multiple symmetries is equal to the figure after translation. (5) translation is determined by direction and distance. (6) after translation, the corresponding line segments are parallel (or collinear) and equal, and the corresponding angles are equal, The line segments connected by the corresponding points are parallel and equal. This kind of moving all the points on the graph at the same distance in a certain direction is called the translational motion of the graph, which is called translational condition for short: the condition for determining a translational motion is the direction and distance of the translation
1. A new figure will be obtained by moving a whole figure along a straight line. The shape and size of the new figure are exactly the same as that of the original figure. 2. Every point in the new figure is obtained by moving a point in the original figure, These two points are corresponding points. The line segments connecting the corresponding points of each group are parallel and equal. 6. The features of Translation: 1. The shape and size of the figure before and after translation remain unchanged, and the position changes. 2. The lines connecting the corresponding points of the new figure and the original figure are parallel and equal. 3. The corresponding line segments of the new figure and the original figure are parallel and equal, and the corresponding angles are equal

(X-2) (xsquare-6x-9) = x (X-5) (x-3)

(X-2) (xsquare-6x-9) = x (X-5) (x-3)
x³-8x²+3x+18=x³-8x²+15x
3x+18=15x
12x=18
x=3/2

If vectors a and B are non-zero vectors, then vector a = - vector B is parallel to vector B___ condition
Why?

For non-zero vectors a and B, your condition is: a = - B, right? A = - B, indicating that a and B have opposite directions, that is, collinear vectors, must be parallel vectors. We can deduce that a ‖ B is a sufficient condition here. If a ‖ B, it only indicates that a and B are collinear, that is, the direction is the same or opposite, and that the relationship between directions does not indicate the relationship between module values

In the triangle ABC, ∠ ACB = 90 degrees, CD ⊥ AB at point D, BF bisection ∠ ABC intersects CD at e, intersects AC at F, prove: CE = CF

∵BC⊥CF,∴∠CFE＝90°－∠CBF.······①
∵ BD ⊥ De, ∫ bed = 90 °～ ABF, obviously: ∫ CEF = ∨ bed, ∨ CEF = 90 °～ ABF,
In addition, CBF = ABF, CEF = 90 ° - ABF
From ① and ②, it can be concluded that: CFE = CEF, CE = CF

The function f (x) = (2-A) (x-1) - 2lnx is known,
Find the extreme value of F (x)!

When a = 1, f (x) = x-2lnx-1
f'(x)=1-2/x
f'(x)=0
1-2/x=0
2/x=1
x=2
f(2)=2-2ln2-1
=1-ln4
The extremum is 1-ln4

If sets A1 and A2 satisfy A1 ∪ A2 = a, then (A1, A2) is a kind of partition of set a, and it is stipulated that if and only if A1 = A2, (A1, A2) and (A2, A1) are the same kind of partition of set a, then what are the different kinds of partition of set a = {1, 2, 3}?

If A1 is a single element set, there are six kinds of partitions, {{1} and {{1} {{{1} and {{2,3}, {1} {{{{{{{{1,2,3}, {2} {{1,2,2,3}, {{3}} {{{1,2,2}, {{3}, {{1,2}}} {{1,2}} {{{{{1,2}} {} {{{} {{{{{} {{{{{{{{{{{and}, {1, 2}, {2, 3} , {1,2,3}; {2,3} and {1}, {1,2}, {1,3}, {1,2,3}; ④ if A1 = a = {1,2,3}, then A2 = ∈, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. To sum up, there are 1 + 6 + 12 + 8 = 27