Live a happy life

Do harm to.

Good harvest, good harvest

Stay calm and keep a clear head.

in good health

Remember/learn/know [something]. memoriter

Hold on to sth.

Hold; Grab/grab/grab

strut about and give oneself airs

summer vacation

go on holiday/vacation

Pay tribute to; in honour of ...

Have high hopes for sb.

Hope to do sth.

to be hospitalized/be admitted to hospital

About an hour or so.

Starve, starve

Go hunting. Go hunting.

In a hurry

I don't know. I don't know

I hope that if ...

Make a good impression on sb.

Move forward slowly and bit by bit.

A friend in need is a true friend. Misfortune tests the sincerity of friends.

Tell sb. something. Inform sb of sth

Insist on doing …

Inspect the factory.

An inspiring speech

Need help

interrupt a conversation

letter of recommendation

Receive an invitation

Invitation letter invitation letter

Tell jokes, tell jokes.

Play a joke on sb.

Travel. Take a trip.

What makes people happy is their own happiness.

Beauty is superficial. Never judge by appearances.

junior school

Just then.

Keep in touch with ...

Keep ... out of ... ...

Key to success

Kick the door

Kick off your shoes. Kick off your shoes.

Kneel/kneel

knock at the door

At the latest

Sooner or later, sooner or later

Burst into laughter.

To violate/abide by the law

Make laws. Make laws

Set the table (for dinner)

Live a simple life

Ignore, omit

Listen to a lecture on … ...

Teach sb. A class

Learn from ... learn from ...

Give a cry of surprise.

Leak news

capital letter

Supine/prone

Come back to life

traffic lights

Make a shopping list.

make a living

Be killed, die; Sacrifice one's life

Frustrated; Lose courage

burst out

Lose a game

Good luck. Good luck.

Washing machine washing machine

be as happy as a lark

mail a package

Make money

Make friends make friends.

Make progress, make progress

Utilize

Make up a story

Make up for mistakes

Be polite. Be polite

A trademark.

full marks

Watch the basketball game.

Hold a contest

I wish you success! I wish you success.

May 1

in this way

By means of ... ...

Never, never

Tailor-made according to the size of.

take measures

Measure height

win gold

Medical team

physical examination

Take medicine. Take some medicine.

Meet the needs of

Encounter a storm

Go to the meeting

have a meeting

have a meeting

in honour of

Be merciless to sb.

Without mercy; Ruthlessly

Under the control of; Be at the mercy of

Merry Christmas! Merry Christmas!

Take a message to sb.

Mid-Autumn Festival

Millions. Millions.

Change your mind. Change your mind.

Be careful of wet paint. Beware of wet paint!

decision

minister of foreign affairs

let the chance slip

Make mistakes, make mistakes.

An error caused by negligence.

Modernity in modernity

change

There is no money on sb.

Choose someone. As a monitor.

One morning,

At the top of the mountain

Join the navy

If necessary

Need help when you need help.

Take on a new look

Hit sb. just in time

take notes

Have no connection with ...

put up a notice

Ignore sb.

In operation

Order sth.

Unemployment unemployment

A pair of glasses a pair of glasses.

The Summer Palace in the Summer Palace

No parking here! No parking here!

Actively participate.

In the past few days, in the past few days.

Be patient with sb.

Practice makes perfect. Practice makes perfect.

Show, show

personally

Take a picture of sb.

play the piano

defloration

Pick up a wallet

Go out for a picnic

A pile of books

Sympathize with sb.

Out of sympathy

Replace, replace

Take sb. Sit in one's seat

Hold, happen

Instead of.

think of a way

Play cards, play cards

Play a joke on sb.

Play with someone.

on the playground

Be satisfied with

Take pleasure in doing sth.

Be rich in life

At what time? ...

Be polite to sb.

Be popular with sb.

occupy

Power station

be in power

Praise sb for sth.

A compliment to ...

attend a meeting

at present

Exchange gifts exchange gifts

Under pressure under pressure

Stop someone. do sth.

At the expense of ... ...

give one’s ears

Be proud of; Be proud of ...

primary school

be put in prison

Serve a sentence in prison

Send sb. Go to prison

break jail

solve problems

answer the question

keep faith

Make a promise, make a promise

happy

Provide food and clothing for the family.

public affairs

Public opinion; popular will

In public, in public.

press

deliberate

Push aside

Knock down, (the wind) blow down

Postpone, postpone

impossible

relay race

By radio

In rags, in rags

At the train station, at the train station

Light rain/heavy rain

a ray of hope

Reach for sth.

be beyond one's power

Be ready to do sth.

In fact, in reality,

Realize one's hopes

For this reason, for this reason.

reception desk

Reference; Speaking of mentioning

Stay in sb's memory.

Remind sb to do sth.

Remind sb. Of sth

At the request of .. ...

Therefore, the result is the result.

Rich; There are a lot of ... Berich came in.

Get rid of

Rob sb. Of something.

Play an important role

Play the role of

Make room

Be rude to sb.

use up

rush hour

Meet someone. Meet sb's needs

Save one's strength

That is to say.

Blame sb. For sth.

Please sit down, please sit down.

Hide something from others.

Grab the thief by the collar. Grab the thief by the collar.

Shake hands with sb.

Shop assistant; salesclerk

Show it to sb. Go out/come in.

show off

Across the bank; On the other side of …

Support sb (political party) on the side of ….

Stand on the side of ...

Out of sight, out of sight

Look, find and catch sight.

Can't see, can't see.

Silent and silent

similar ...

One-way ticket one-way ticket

Take the size of.

Give someone a note quietly.

Slip a note into someone's hand.

Slip on the snow

Overcome difficulties, overcome difficulties

About so.

in connection with

National anthem

Speak boldly, clearly and loudly.

make a speech

At the speed of ... ...

square kilometer

Support, support

starve to death

in good condition

brick by brick

keep faith

Lie on the ground, prone.

A four-story house.

Catch up with the storm and be trapped by the storm

Be strict with sb. In sth.

strike a match

Struggling to stand up

Study it carefully.

Suddenly, suddenly.

summer vacation

Supply sb. Use sth.

To sb's surprise.

Wipe the sweat off your face.

Sit down and eat. Sit down and eat.

pay tax

make tea

Through telescope

Tell a story

Distinguish between the two

Take your temperature.

Tens of thousands of tens of thousands.

Be frightened by ... ...

Thank sb. For something.

throw away

Spit out (food), vomit

Soon.

traffic jam

Play a trick on sb.

Be in trouble (distress)

a pair of trousers

go to college

Visit sb.

At the top of my lungs.

In the war, in the war

Wear; exhaust

Pull up weeds

Dressed in white, dressed in white

Generally speaking, as a whole.

On the whole.

Where there is a will, there is a way.

Where there is a will, there is a way.

Be willing to do sth.

Wipe off the dust, wipe off the dust.

work wonders

Not surprising; No wonder no wonder

interpose

Have a word with someone.

In short, in a word

Math:

(√ is the root sign)

1 There is only one straight line at two points.

The line segment between two points is the shortest.

The complementary angles of the same angle or equal angle are equal.

The complementary angles of the same angle or the same angle are equal.

One and only one straight line is perpendicular to the known straight line.

Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.

7 Parallel axiom passes through a point outside a straight line, and there is only one straight line parallel to this straight line.

If both lines are parallel to the third line, the two lines are also parallel to each other.

The same angle is equal and two straight lines are parallel.

The internal dislocation angles of 10 are equal, and the two straight lines are parallel.

1 1 are complementary and two straight lines are parallel.

12 Two straight lines are parallel and have the same angle.

13 two straight lines are parallel, and the internal dislocation angles are equal.

14 Two straight lines are parallel and complementary.

Theorem 15 The sum of two sides of a triangle is greater than the third side.

16 infers that the difference between two sides of a triangle is smaller than the third side.

The sum of the internal angles of 17 triangle is equal to 180.

18 infers that the two acute angles of 1 right triangle are complementary.

19 Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

Inference 3 The outer angle of a triangle is greater than any inner angle that is not adjacent to it.

2 1 congruent triangles has equal sides and angles.

Axiom of Angular (SAS) has two triangles with equal angles.

The Axiom of 23 Angles (ASA) has the congruence of two triangles, which have two angles and their sides correspond to each other.

The inference (AAS) has two angles, and the opposite side of one angle corresponds to the congruence of two triangles.

The axiom of 25 sides (SSS) has two triangles with equal sides.

Axiom of hypotenuse and right angle (HL) Two right angle triangles with hypotenuse and right angle are congruent.

Theorem 1 The distance between a point on the bisector of an angle and both sides of the angle is equal.

Theorem 2 is a point with equal distance on both sides of an angle, which is on the bisector of this angle.

The bisector of an angle 29 is the set of all points with equal distance to both sides of the angle.

The nature theorem of isosceles triangle 30 The two base angles of isosceles triangle are equal (that is, equilateral and equiangular).

3 1 Inference 1 The bisector of the vertices of an isosceles triangle bisects the base and is perpendicular to the base.

The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.

Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.

34 Judgment Theorem of an isosceles triangle If a triangle has two equal angles, then the opposite sides of the two angles are also equal (equal angles and equal sides).

Inference 1 A triangle with three equal angles is an equilateral triangle.

Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle.

In a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.

The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.

Theorem 39 The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal.

The inverse theorem and the point where the two endpoints of a line segment are equidistant are on the middle vertical line of this line segment.

The perpendicular bisector of the 4 1 line segment can be regarded as the set of all points with equal distance from both ends of the line segment.

Theorem 42 1 Two graphs symmetric about a line are conformal.

Theorem 2: If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular to the straight line connecting the corresponding points.

Theorem 3 Two graphs are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.

45 Inverse Theorem If the straight line connecting the corresponding points of two graphs is bisected vertically by the same straight line, then the two graphs are symmetrical about this straight line.

46 Pythagorean Theorem The sum of squares of two right angles A and B of a right triangle is equal to the square of the hypotenuse C, that is, A 2+B 2 = C 2.

47 Inverse Theorem of Pythagorean Theorem If the three sides of a triangle A, B and C are related in length A 2+B 2 = C 2, then the triangle is a right triangle.

The sum of the quadrilateral internal angles of Theorem 48 is equal to 360.

The sum of the external angles of the quadrilateral is equal to 360.

The theorem of the sum of internal angles of 50 polygons is that the sum of internal angles of n polygons is equal to (n-2) × 180.

5 1 It is inferred that the sum of the external angles of any polygon is equal to 360.

52 parallelogram property theorem 1 parallelogram diagonal equality

53 parallelogram property theorem 2 The opposite sides of parallelogram are equal

It is inferred that the parallel segments sandwiched between two parallel lines are equal.

55 parallelogram property theorem 3 diagonal bisection of parallelogram.

56 parallelogram decision theorem 1 Two groups of parallelograms with equal diagonals are parallelograms.

57 parallelogram decision theorem 2 Two groups of parallelograms with equal opposite sides are parallelograms.

58 parallelogram decision theorem 3 A quadrilateral whose diagonal is bisected is a parallelogram.

59 parallelogram decision theorem 4 A group of parallelograms with equal opposite sides are parallelograms.

60 Rectangle Property Theorem 1 All four corners of a rectangle are right angles.

6 1 rectangle property theorem 2 The diagonals of rectangles are equal

62 Rectangular Decision Theorem 1 A quadrilateral with three right angles is a rectangle.

63 Rectangular Decision Theorem 2 Parallelograms with equal diagonals are rectangles

64 diamond property theorem 1 all four sides of the diamond are equal.

65 Diamond Property Theorem 2 Diagonal lines of diamonds are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.

66 Diamond area = half of diagonal product, that is, S=(a×b)÷2.

67 diamond decision theorem 1 A quadrilateral with four equilateral sides is a diamond.

68 Diamond Decision Theorem 2 Parallelograms whose diagonals are perpendicular to each other are diamonds.

69 Theorem of Square Properties 1 All four corners of a square are right angles and all four sides are equal.

Theorem of 70 Square Properties 2 Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines.

Theorem 7 1 1 is congruent with respect to two centrosymmetric graphs.

Theorem 2 About two graphs with central symmetry, the connecting lines of symmetric points both pass through the symmetric center and are equally divided by the symmetric center.

Inverse Theorem If the corresponding points of two graphs pass through a certain point and are connected by it.

If the point is split in two, then the two graphs are symmetrical about the point.

The property theorem of isosceles trapezoid is that two angles of isosceles trapezoid on the same base are equal.

The two diagonals of an isosceles trapezoid are equal.

76 isosceles trapezoid decision theorem A trapezoid with two equal angles on the same base is an isosceles trapezoid.

A trapezoid with equal diagonal lines is an isosceles trapezoid.

Theorem of bisecting line segments by parallel lines If a group of parallel lines are tangent to a straight line.

Equal, then the line segments cut on other straight lines are also equal.

79 Inference 1 A straight line passing through the midpoint of one waist of a trapezoid and parallel to the bottom will bisect the other waist.

Inference 2 A straight line passing through the midpoint of one side of a triangle and parallel to the other side will be equally divided.

Trilaterality

The median line theorem of 8 1 triangle The median line of a triangle is parallel to and equal to the third side.

Half of

The trapezoid midline theorem is parallel to the two bottoms and equals the sum of the two bottoms.

Half l = (a+b) ÷ 2s = l× h。

Basic properties of ratio 83 (1) If a:b=c:d, then ad=bc.

If ad=bc, then a: b = c: d.

84 (2) Combinatorial Properties If A/B = C/D, then (A B)/B = (C D)/D.

85 (3) Isometric Property If A/B = C/D = … = M/N (B+D+…+N ≠ 0), then

(a+c+…+m)/(b+d+…+n)=a/b

86 parallel lines are divided into segments and proportional theorems. Three parallel lines cut two straight lines and get the corresponding results.

The line segments are proportional.

It is inferred that the line parallel to one side of the triangle cuts the other two sides (or the extension lines of both sides), and the corresponding line segments are proportional.

Theorem 88 If the corresponding line segments obtained by cutting two sides (or extension lines of two sides) of a triangle are proportional, then this straight line is parallel to the third side of the triangle.

A straight line parallel to one side of a triangle and intersecting with the other two sides, the three sides of the cut triangle are directly proportional to the three sides of the original triangle.

Theorem 90 A straight line parallel to one side of a triangle intersects the other two sides (or extension lines of both sides), and the triangle formed is similar to the original triangle.

9 1 similar triangles's decision theorem 1 Two angles are equal and two triangles are similar (ASA)

Two right triangles divided by the height on the hypotenuse are similar to the original triangle.

Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).

Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS)

Theorem 95 If the hypotenuse of a right triangle and one right-angled side and another right-angled side

The hypotenuse of an angle is proportional to a right-angled side, so two right-angled triangles are similar.

96 Property Theorem 1 similar triangles has a high ratio, and the ratio corresponding to the center line is flat with the corresponding angle.

The ratio of dividing lines is equal to the similarity ratio.

97 Property Theorem 2 The ratio of similar triangles perimeter is equal to similarity ratio.

98 Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.

The sine value of any acute angle is equal to the cosine value of other angles, the cosine value of any acute angle, etc.

Sine value of other angles

100 The tangent of any acute angle is equal to the cotangent of other angles, the cotangent of any acute angle, etc.

Tangent value of its complementary angle

10 1 A circle is a set of points whose distance from a fixed point is equal to a fixed length.

102 The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.

The outer circle of 103 circle can be regarded as a collection of points whose center distance is greater than the radius.

104 The radius of the same circle or equal circle is the same.

105 The distance from the fixed point is equal to the trajectory of the fixed point, with the fixed point as the center, and the fixed length is half.

Diameter circle

106 and it is known that the locus of the point with the same distance between the two endpoints of the line segment is perpendicular to the line segment.

bisector

The locus from 107 to a point with equal distance on both sides of a known angle is the bisector of this angle.

The trajectory from 108 to the point with the same distance from two parallel lines is parallel to these two parallel lines with a distance of.

A straight line of equality

Theorem 109 Three points that are not on the same straight line determine a circle.

1 10 vertical diameter theorem divides the chord perpendicular to its diameter into two parts, and divides the two arcs opposite to the chord into two parts.

1 1 1 inference 1 ① bisect the diameter of the chord (not the diameter) perpendicular to the chord and bisect the two arcs opposite the chord.

(2) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord.

③ bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.

1 12 Inference 2 The arcs sandwiched by two parallel chords of a circle are equal.

1 13 circle is a centrosymmetric figure with the center of the circle as the symmetry center.

Theorem 1 14 In the same circle or in the same circle, arcs with equal central angles are equal, and chords with equal central angles are equal.

Equal, the chord center distance of the opposite chord is equal.

1 15 inference in the same circle or in the same circle, if two central angles, two arcs, two chords or two.

If one set of quantities in the chord-to-chord distance is equal, then the other sets of quantities corresponding to it are also equal.

Theorem 1 16 The angle of an arc is equal to half its central angle.

1 17 Inference 1 The circumferential angles of the same arc or the same arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

1 18 Inference 2 The circumferential angle (or diameter) of a semicircle is a right angle; 90 degree circle angle

The chord on the right is the diameter.

1 19 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.

120 Theorem The inscribed quadrilateral of a circle is diagonally complementary, and any external angle is equal to it.

Internal diagonal of

12 1① the intersection of the straight line l and ⊙O is d < r.

(2) the tangent of the straight line l, and ⊙ o d = r.

③ lines l and ⊙O are separated by d > r.

122 tangent theorem The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.

123 The property theorem of tangent line The tangent line of a circle is perpendicular to the radius passing through the tangent point.

124 Inference 1 A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.

125 Inference 2 A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.

126 tangent length theorem leads to two tangents of a circle from a point outside the circle, and their tangent lengths are equal.

The line between the center of the circle and this point bisects the included angle between the two tangents.

127 The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.

128 Chord Angle Theorem The chord angle is equal to the circumferential angle of the arc pair it clamps.

129 Inference: If the arc enclosed by two chord tangent angles is equal, then the two chord tangent angles are also equal.

130 intersection chord theorem The product of two intersecting chords in a circle divided by the intersection point.

(to) equal to ...

13 1 Inference: If the chord intersects the diameter vertically, then half of the chord is formed by dividing it by the diameter.

Proportional median of two line segments

132 tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the tangent length is the point to be cut.

The proportional average of the lengths of two straight lines at the intersection of a straight line and a circle.

133 It is inferred that two secant lines of the circle are drawn 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 line and the circle is equal.

134 If two circles are tangent, then the tangent point must be on the line.

135① perimeter of two circles D > R+R ② perimeter of two circles d = r+r.

③ the intersection of two circles r-r < d < r+r (r > r).

④ inscribed circle D = R-R (R > R) ⑤ two circles contain D < R-R (R > R).

Theorem 136 The intersection of two circles bisects the common chord of two circles vertically.

Theorem 137 divides a circle into n (n ≥ 3);

(1) The polygon obtained by connecting the points in turn is the inscribed regular N polygon of this circle.

(2) The tangent of a circle passing through each point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular N polygon of the circle.

Theorem 138 Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.

139 every inner angle of a regular n-polygon is equal to (n-2) ×180/n.

140 Theorem Radius and apothem Divides a regular N-polygon into 2n congruent right triangles.

14 1 the area of the regular n polygon Sn = PNRN/2 P represents the perimeter of the regular n polygon.

142 The area of a regular triangle √ 3a/4a indicates the side length.

143 if there are k positive n corners around a vertex, then the sum of these angles should be

360, so k× (n-2) 180/n = 360 is changed to (n-2)(k-2)=4.

The formula for calculating the arc length of 144 is L = NR/ 180.

145 sector area formula: s sector =n r 2/360 = LR/2.

146 inner common tangent length = d-(R-r) outer common tangent length = d-(R+r)

There are still some, please help to supplement them. )

Practical tools: common mathematical formulas

Formula classification formula expression

Multiplication and factorization A2-B2 = (a+b) (a-b) A3+B3 = (a+b) (A2-AB+B2) A3-B3 = (A-B (A2+AB+B2))

Trigonometric inequality | A+B |≤| A |+B||||| A-B|≤| A |+B || A |≤ B < = > -b≤a≤b

|a-b|≥|a|-|b| -|a|≤a≤|a|

The solution of the unary quadratic equation -b+√(b2-4ac)/2a -b-√(b2-4ac)/2a

The relationship between root and coefficient x1+x2 =-b/ax1* x2 = c/a Note: Vieta theorem.

discriminant

B2-4ac=0 Note: This equation has two equal real roots.

B2-4ac >0 Note: The equation has two unequal real roots.

B2-4ac & lt； Note: The equation has no real root, but a complex number of the yoke.

Conical volume formula V= 1/3*S*H Conical volume formula V= 1/3*pi*r2h