a handful of
live a happy life
damage, harm do harm to
a good harvest a good harvest
keep one's head
in good health
remember something learn / know sth. by heart
Keep a tight hold on sth.
Hold; catch / take / get hold of
Hold one's head high
the summer holidays
on holiday
to pay respect to (to)...; to commemorate... in honour of
to have great hopes for someone have high hope for sb.
in the hope of doing sth.
be in hospital
an hour or so
go hungry
go hunting
in a hurry
have no idea
If only...if only
make a good impression on sb.
inch one's way forward
A friend in need is a true friend. A friend in need is a friend indeed.
Inform sb of sth
insist on doing
inspect a factory
an inspiring speech
in instant need of help
interrupt a conversation
Introduction letter a letter of introduction
receive an invitation
a letter of invitation
tell a joke
Play a joke with sb.
Travel make a journey
What makes someone happy is to one's joy
Don't judge a book by its cover.
Don't judge a man by his looks.
a junior high school
just then
keep in touch with
keep out of
the key to success
kick the door
kick off your shoes kick off one's shoes
go down / fall on one's knees
knock at the door
at the latest, at the latest
p>
sooner or later
burst into laughter
break / obey the law
make a law make a law
lay the table
lead a simple life
ignore, omit leave out
attend a lecture on
teach sb. a lesson
take a lesson from
let out a cry of surprise
let out the news
a capital letter
Lie on your back / prone lie on one's back / stomach
come back to life
traffic lights
make a shopping list
make a shopping list
p>
make a living, make a living
lose one's life
lose one's life
lose heart
lose one's voice
lose a game
Wish you good luck.
Wish you good luck.
a washing machine
be mad with joy
send the parcel by mail
make money
make friends
make progress
make use of
make up a story
make up for one's mistake
have good manners
a trade mark
full marks
watch a basketball match
have a match
I wish you success. May you succeed.
May Day
by this means
by means of
Never by no means
make...to one's measure
take a measure
measure one's height
get a gold medal
a medical team
medical examination
take / have some medicine
p>
meet the needs of
meet with a storm
go to a meeting
have a meeting meeting
hold a meeting
in memory of
have no mercy on sb.
Without mercy; cruelly without mercy
Under the control of... at the mercy of
Merry Christmas! Merry Christmas!
Take a message for sb.
Mid-autumn Day
Millions, many, many Millions of
change one's mind
Beware of wet paint.
Mind the wet paint!
make up one's mind
the minister of foreign affairs
miss an opportunity
make a mistake
caused by negligence
in modern times
small money
on someone have no money with sb.
choose someone to be the monitor make sb. monitor
on the early morning
on the top of the mountain at the top of the mountain
join the navy
if necessary if necessary
in need of help
take on a new look
hit sb. on the nose
make / take notes
with… have nothing to do with
put up a notice
pay no notice to sb.
operating, in implementation be in operation
order something place an order for sth.
out of work
a pair of glasses
The Summer Palace
No parking here! No parking here!
take an active part in
in the past few days
to someone be patient with sb.
Practice makes perfect.
Practice makes perfect.
Perform, put on performances
In person, in person
Take a photo of sb.
play the piano
pick flowers
pick up a wallet
go out for a picnic
a pile of books
pity someone (help someone out of sympathy) have / take pity on sb.
out of pity
Replace in place of
Sit in someone's seat, replace someone's position, take one's place
Hold, take place
Replace, agent take the place of
make a plan
play cards
play a joke on sb.
play with sb.
on the playground
be pleased with
like to do something take pleasure in doing sth.
live in plenty
on the point of
be polite to sb .
be popular with sb.
possession, possess take possession of
power station
be in power, take power
praise someone for something praise sb. for sth.
praise in praise of
be present at a meeting
at present
exchange presents
under pressure
prevent someone from doing something prevent sb. from doing
At the price of
No matter how much it costs (at any price) at any price
Be proud of; feel proud of taking pride in
primary school
go to prison
be in prison
will someone throw / put sb. into prison
escape from prison
solve the problem
answer the question
keep one's promise
>
Promise, make a promisemake a promise
be proud of
provide food and clothes for one's family
公* ** Affairs public affairs
Public opinion
In public, in public
Publishing house
Deliberately on purpose
Push aside
Push over, push over
Delay, postpone put off
Impossible out of the question
a relay race
on the radio
rags in rags
at the railway station
light / heavy rain
a ray of hope
reach for sth .
out of ones' reach
willing to do something be ready to do
actually in reality
fulfill hope realize one's hope
for this reason
reception desk
reference; refer to
remain in one's memory
remind someone to do something remind sb. to do sth.
remind someone of remind sb. of sth.
p>
by request
As a result
be rich in
get rid of of
rob someone or something rob sb. of sth.
play an important role
play the role of... play the role of
make room for...
be rude to sb.
run out of
rush hour
satisfy one's needs
save one's strength
that is to say
p>To blame someone for something scold sb. for sth.
Sit down, take one's seat
Don't let others know about something, keep it secret keep sth. a secret
seize a thief by the collar
shake hands with sb.
p>
Shop assistant; salesperson shop assistant
Take someone out/in show sb. out / in
Show off show off
The other side; on the other side... on the other side of
support someone take the side of
stand on the other side of
lose sight of
See, catch sight out
Out of sight
Quietly in silence
Similar to be similar to
single ticket
take the size of
secretly slip a note to someone
slip a note into one's hand
slip on the snow
overcome difficulties smooth away difficulties
about or so
p>have something to do with
the national song
speak boldly, speak out clearly and loudly
make a speech, make a speech
at a speed of
square kilometers square kilometers
represent, symbolize stand for
starve to death
in a good state
step by step
stick to one's word / promise
lie on one's stomach
a house of four storeys
be caught in the storm
be strict with sb. in sth.
strike a match
struggle to one's feet
carefully make a study of
all of a sudden
summer holidays
supply sb. with sth.
p>
To one's surprise
sweat off one's face
sit down to table
pay one's taxes
make tea
through a telescope
tell a story
distinguish, distinguish tell one from the other
take one's temperature
tens of thousands of
be terrified at
give thanks for something someone be thankful to sb. for sth.
throw away
vomit (food), vomit throw up
immediately, soon in no time
Traffic jam
Play a trick on someone, trick someone.
Be in trouble
Be in trouble
p>
a pair of trousers
attend university
pay a visit to sb.
loudly (shout) ) at the top of one's voice
at war
wear out; wear out
pull out the weeds
be dressed in white
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
make wonders
No wonder; no wonder no wonder
get in a word
Speak to someone have a word with sb.
In short, in a word
Mathematics:
····(√ is the root sign)
1 There is only one straight line through two points
2 The shortest line segment between two points
3 The supplementary angles of congruent or equal angles are equal
4 The supplementary angles of congruent angles or equal angles are equal
5 There is and is only one straight line perpendicular to the known straight line through a point
6 All line segments connecting a point outside the straight line and each point on the straight line In , the perpendicular segment is the shortest
7 Parallel axiom passes through a point outside the straight line, and there is only one straight line parallel to this straight line
8 If two straight lines are parallel to the third straight line, These two straight lines are also parallel to each other
9 If the concentric angles are equal, the two straight lines are parallel
10 If the interior angles are equal, the two straight lines are parallel
11 The interior angles on the same side are complementary , two straight lines are parallel
12 Two straight lines are parallel and congruent angles are equal
13 Two straight lines are parallel and interior angles are equal
14 Two straight lines are parallel and interior angles on the same side are equal Complementarity
15 Theorem The sum of the two sides of a triangle is greater than the third side
16 It follows that the difference between the two sides of a triangle is less than the third side
17 The sum of the angles of a triangle is three The sum of the interior angles is equal to 180°
18 Corollary 1 The two acute angles of a right triangle are complementary
19 Corollary 2 An exterior angle of a triangle is equal to the sum of two interior angles that are not adjacent to it
20 Corollary 3 An exterior angle of a triangle is greater than any interior angle that is not adjacent to it
21 The corresponding sides and corresponding angles of congruent triangles are equal
22 Side-Angle-Side Axiom (SAS) Two triangles are congruent if there are two sides and their included angles are equal
23 Angle-Side Axiom (ASA) Two triangles are congruent if there are two angles and their included sides are equal. Triangles Congruent
24 Corollary (AAS) Two triangles are congruent if they have two angles and the opposite sides of one of the angles are equal
25 Side-Side Axiom (SSS) There are three sides corresponding Two equal triangles are congruent
26 Hypotenuse and right-angled side axiom (HL) Two equal right-angled triangles with hypotenuse and a right-angled side are congruent
27 Theorem 1 The distance from a point on the bisector of an angle to both sides of the angle is equal
28 Theorem 2 A point that is the same distance from both sides of an angle is on the bisector of the angle
31 Corollary 1 The bisector of the vertex of an isosceles triangle bisects the base and is perpendicular to the base
32 The bisector of the vertex of an isosceles triangle, the midline on the base and the base The heights above coincide with each other
33 Corollary 3 The angles of an equilateral triangle are equal, and each angle is equal to 60°
34 The determination theorem of an isosceles triangle If a triangle has If two angles are equal, then the sides opposite the two angles are also equal (equal angles to equal sides)
35 Corollary 1 A triangle with three equal angles is an equilateral triangle <
/p>
36 Corollary 2 An isosceles triangle with an angle equal to 60° is an equilateral triangle
37 In a right-angled triangle, if an acute angle is equal to 30°, then the right-angled side it opposes is equal to Half of the hypotenuse
38 The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse
39 Theorem The distance between a point on the perpendicular bisector of a line segment and the two endpoints of the line segment Equality
40 The converse theorem and the point at which the distance between the two endpoints of a line segment are equal is on the perpendicular bisector of the line segment
41 The perpendicular bisector of the line segment can be regarded as two The set of all points whose endpoints are equidistant
42 Theorem 1 Two figures that are symmetrical about a certain straight line are congruent
43 Theorem 2 If two figures are symmetrical about a certain straight line, Then the axis of symmetry is the perpendicular bisector of the line connecting the corresponding points
44 Theorem 3 Two figures are symmetrical about a straight line. If their corresponding line segments or extended lines intersect, then the intersection point is on the axis of symmetry
45 Converse Theorem If the line connecting the corresponding points of two figures is perpendicularly bisected by the same straight line, then the two figures are symmetrical about this straight line
46 Pythagorean Theorem The two right-angled sides of a right triangle are a, The sum of the squares of b is equal to the square of the hypotenuse c, that is, a^2 b^2=c^2
47 The converse theorem of the Pythagorean theorem. If the lengths a, b, and c of the three sides of the triangle are related a^2 b^2=c^2, then this triangle is a right triangle
48 Theorem The sum of the interior angles of a quadrilateral is equal to 360°
49 The sum of the exterior angles of a quadrilateral is equal to 360°
p>
50 The sum of the interior angles of a polygon theorem The sum of the interior angles of an n-sided polygon is equal to (n-2) × 180°
51 The sum of the exterior angles of any polygon is equal to 360°
52 Theorem of properties of parallelograms 1 The opposite angles of parallelograms are equal
53 Theorem of properties of parallelograms 2 The opposite sides of parallelograms are equal
54 Deduction of parallel line segments sandwiched between two parallel lines Equality
55 Parallelogram Properties Theorem 3 The diagonals of a parallelogram bisect each other
56 Parallelogram Determination Theorem 1 A quadrilateral whose two sets of diagonals are equal is a parallelogram
57 Parallelogram Determination Theorem 2 A quadrilateral whose two opposite sides are equal is a parallelogram
58 Parallelogram Determination Theorem 3 A quadrilateral whose diagonals bisect each other is a parallelogram
59 Parallelogram Determination Theorem 4 A set of parallelograms with equal opposite sides is a parallelogram
60 Rectangle Properties Theorem 1 The four corners of a rectangle are right angles
61 Rectangle Properties Theorem 2 Rectangle The diagonals are equal
62 Rectangle Determination Theorem 1 A quadrilateral with three right angles is a rectangle
63 Rectangle Determination Theorem 2 A parallelogram with equal diagonals is a rectangle
64 Rhombus Property Theorem 1 The four sides of a rhombus are equal
65 Rhombus Property Theorem 2 The diagonals of a rhombus are perpendicular to each other, and each diagonal bisects a set of diagonals
66 The area of ??a rhombus = half the product of the diagonals, that is, S = (a × b) ÷ 2
67 The rhombus determination theorem 1 A quadrilateral with all four sides equal is a rhombus
68 Rhombus Determination Theorem 2 A parallelogram with perpendicular diagonals is a rhombus
69 Square Properties Theorem 1 The four angles of a square are all right angles and all four sides are equal
70 Square Properties Theorem 2 The two diagonals of a square are equal and bisect each other perpendicularly. Each diagonal bisects a set of diagonals
71 Theorem 1 Two figures that are symmetric about the center are congruent
72 Theorem 2 Regarding two centrally symmetric figures, the lines connecting the symmetry points pass through the symmetry center and are bisected by the symmetry center
73 The converse theorem: If the lines connecting the corresponding points of the two figures both pass through
passes a certain point and is bisected by this point
Then the two figures are symmetrical about this point
74 Theorem of Properties of Isosceles Trapezoid Two isosceles trapezoids on the same base Angle equal
75 The two diagonals of an isosceles trapezoid are equal
76 Isosceles trapezoid determination theorem A trapezoid with two equal angles on the same base is an isosceles trapezoid
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77 A trapezoid with equal diagonals is an isosceles trapezoid
78 Parallel lines bisect the segment theorem If a set of parallel lines intercepts a line segment on a straight line
If equal, then the line segments intercepted on other straight lines are also equal
79 Corollary 1 A straight line passing through the midpoint of one waist of the trapezoid and parallel to the base will bisect the other waist
80 Corollary 2 A straight line passing through the midpoint of one side of a triangle and parallel to the other side must bisect the third side
81 Median line theorem of a triangle The median line of a triangle is parallel to the third side, And equal to half of it
82 Trapezoid median line theorem The median line of a trapezoid is parallel to the two bases, and equal to
half of the sum of the two bases L = (a b)÷2 S=L×h
83 (1) Basic properties of proportion If a: b=c: d, then ad=bc
If ad=bc , then a: b=c: d
84 (2) Composite property If a/b=c/d, then (a±b)/b=(c±d)/d
85 (3) Proportional property If a/b=c/d=…=m/n(b d… n≠0), then
(a c… m)/(b d … n)=a／b
86 Theorem of Proportional Line Segments of Parallel Lines If three parallel lines cut two straight lines, the corresponding corresponding
line segments are proportional
87 Inference: If a straight line parallel to one side of a triangle cuts the other two sides (or the extensions of both sides), the corresponding line segments obtained are proportional
88 Theorem If a straight line cuts both sides of the triangle (or the extensions of both sides), the obtained The corresponding line segments are proportional, then this straight line is parallel to the third side of the triangle
89 A straight line parallel to one side of the triangle and intersecting the other two sides, the three sides of the triangle intercepted are the same as the three sides of the original triangle The sides are proportional
90 Theorem: If a straight line parallel to one side of a triangle intersects the other two sides (or extensions of both sides), the triangle formed is similar to the original triangle
91 Judgment of similar triangles Theorem 1 The two angles are equal and the two triangles are similar (ASA)
92 The two right triangles divided by the height of the hypotenuse are similar to the original triangle
93 Determination Theorem 2 If the two sides are proportional and the angles are equal, the two triangles are similar (SAS)
94 Determination Theorem 3 If the three sides are proportional, the two triangles are similar (SSS)
95 Theorem If a right angle The hypotenuse and a right-angled side of a triangle are proportional to the hypotenuse and a right-angled side of another right triangle
Then the two right-angled triangles are similar
96 Property Theorem 1 The ratio of the corresponding heights of similar triangles, the ratio of the corresponding midlines and the corresponding angles
The ratios of the bisection lines are all equal to the similarity ratio
97 Property Theorem 2 The ratio of the perimeters of similar triangles is equal to the similarity ratio
98 Property Theorem 3 The ratio of the areas of similar triangles is equal to the square of the similarity ratio
99 The sine value of any acute angle is equal to the cosine value of its complementary angle, the cosine value of any acute angle, etc.
The sine of its supplementary angle
100 The tangent of any acute angle is equal to the cotangent of its supplementary angle, the cotangent of any acute angle, etc.
The tangent of its complementary angle
101 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 circle
The set of points whose center distance is less than the radius
103 The outside of a circle can be regarded as the set of points whose center distance is greater than the radius
104 The radii of congruent or equal circles are equal
The trajectory of a point whose distance from 105 to the fixed point is equal to the fixed length is a circle with the fixed point as the center and the fixed length as the semi-diameter
106 and the known The locus of a point that is equidistant from the two endpoints of a line segment is the perpendicular
bisector of the line segment
107. The locus of a point that is equidistant from both sides of a known angle is this The locus of the bisector of an angle
108 to a point equidistant from two parallel lines is a straight line parallel to and equidistant
from the two parallel lines
Theorem 109 does not determine a circle from three points on the same straight line.
110 Perpendicular Diameter Theorem The diameter of a string perpendicular to the string bisects the string and bisects the two arcs subtended by the string
111 Corollary 1 ①The diameter of the bisected chord (not the diameter) is perpendicular to The chord, and bisects the two arcs subtended by the chord
②The perpendicular bisector of the chord passes through the center of the circle, and bisects the two arcs subtended by the chord
③Bisectors the one subtended by the chord The diameter of the arc bisects the chord perpendicularly and bisects the other arc subtended by the chord
112 Corollary 2 The arcs between two parallel chords of a circle are equal
113 The center of a circle is Theorem 114 is a centrosymmetric figure with the center of symmetry
Theorem: In identical circles or equal circles, the arcs subtended by equal central angles are equal, and the chords subtended by them are equal.
Theorems subtended by The chord-center distances of strings are equal
115 Corollary In the same circle or equal circles, if the central angles of two circles, two arcs, two chords or the chord-center distances of two strings
If a set of quantities in is equal, then the remaining sets of quantities corresponding to them are equal
116 Theorem The circumferential angle subtended by an arc is equal to half of the central angle subtended by it
117 Corollary 1 The circumferential angles subtended by the same arc or equal arcs are equal; in the same circle or equal circles, the arcs subtended by equal circumferential angles are also equal
118 Corollary 2 The circumferential angles subtended by a semicircle (or diameter) is a right angle; the chord subtended by a circumferential angle of 90° is the diameter
119 Corollary 3 If the midline on one side of a triangle is equal to half of this side, then the triangle is a right triangle
120 Theorem The diagonal angles of an inscribed quadrilateral of a circle are complementary, and any external angle is equal to its
internal opposite angle
121 ① Line L intersects ⊙O d＜r
②The straight line L and ⊙O are tangent d=r
③The straight line L and ⊙O are separated d>r
122 Determination of tangent line Theorem A straight line that passes through the outer end of a radius and is perpendicular to this radius is a tangent to a circle
123 Properties of tangents Theorem The tangent of a circle is perpendicular to the radius passing through the tangent point
124 Corollary 1 A straight line that passes through the center of a circle and is perpendicular to the tangent line must pass through the tangent point
125 Corollary 2 A straight line that passes through the tangent point and is perpendicular to the tangent line must pass through the center of the circle
126 The tangent length theorem leads from a point outside the circle The two tangent lines of a circle are equal in length.
The line connecting the center of the circle and this point bisects the angle between the two tangent lines.
127 Two pairs of quadrilaterals circumscribing a circle The sum of the sides is equal
128 Chordal tangent angle theorem The chordal tangent angle is equal to the circumferential angle of the arc pair it contains
129 Corollary If the arcs enclosed by two chordal tangent angles are equal, Then the tangent angles of these two chords are also equal
130 Intersecting Chord Theorem For two intersecting chords in a circle, the products of the lengths of the two line segments divided by the intersection points are equal
Equal
131 Corollary: If a chord intersects the diameter perpendicularly, then half of the chord is divided into diameters
The median of the ratio of two line segments
132 Cutting line theorem from the circle The tangent and secant lines of a circle are drawn from an external point. The length of the tangent is the middle term of the ratio of the lengths of the two line segments from this point to the intersection of the secant
line and the circle
133 Inference from a point outside the circle The two secants leading a circle, the products of the lengths of the two line segments from this point to the intersection of each secant and the circle are equal
134 If two circles are tangent, then the tangent point must be on the center line
135 ① The two circles are circumscribed by d＞R r ② The two circles are circumscribed by d=R r
③The two circles intersect R-r＜d＜R r(R＞r)
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④Two circles are inscribed d=R-r(R＞r) ⑤Two circles are inscribed d＜R-r(R＞r)
136 Theorem The line connecting the centers of two intersecting circles bisects them perpendicularly The common chord of a circle
The 137 theorem divides the circle into n (n≥3):
⑴The polygon obtained by connecting the points in sequence is the inscribed positive n of the circle Polygon
⑵ Draw tangents to the circle through each point, and the polygon with the intersection of adjacent tangents as the vertex is a regular n-gon circumscribed to the circle
138 Theorem Any regular polygon has a circumscribed circle and an inscribed circle, these two circles are concentric circles
139 Each interior angle of a regular n-sided polygon is equal to (n-2) × 180°/n
140 Theorem The radius and side center distance of the regular n-sided polygon divide the regular n-sided polygon into 2n congruent right triangles
141 The area of ??the regular n-sided polygon Sn=pnrn/2 p represents the perimeter of the regular n-sided polygon Length
142 The area of ??an equilateral triangle √3a/4 a represents the side length
143 If there are k angles of a regular n-sided polygon around a vertex, the sum of these angles should be
360°, so k×(n-2)180°/n=360° becomes (n-2)(k-2)=4
144 arc length calculation Formula: L=n兀R／180
145 sector area formula: S sector=n兀R^2／360=LR／2
146 inner common tangent length = d- (R-r) Grandfather’s tangent length = d-(R r)
(There are some more, please help to add)
Practical tools: commonly used mathematical formulas
Formula classification formula expression
Multiplication and factoring a2-b2=(a b)(a-b) a3 b3=(a b)(a2-ab b2) a3-b3=(a-b(a2 ab b2)
Triangle inequality |a b|≤|a| |b| |a-b|≤|a| |b| |a|≤blt;=gt;-b≤a≤b
|a-b|≥|a|-|b| -|a|≤a≤|a|
Solution of quadratic equation-b √(b2-4ac)/2a -b-√( b2-4ac)/2a
The relationship between roots and coefficients X1 X2=-b/a X1*X2=c/a Note: Vedic Theorem
Discriminant
b2-4ac=0 Note: The equation has two equal real roots
b2-4acgt;0 Note: The equation has two unequal real roots
b2- 4aclt; 0 Note: The equation has no real roots, but ***yoke complex roots
Cone volume formula V=1/3*S*H Cone volume formula V=1/3*pi*r2h