For any triangle with three sides A, B and C, a triangle with A, B and C satisfies the property.
(Note: a*b and a*c are respectively A times B and A times C C..A 2, B 2 and C 2 are the squares of A, B and C. )
a^2=b^2+c^2-2*b*c*CosA
b^2=a^2+c^2-2*a*c*CosB
c^2=a^2+b^2-2*a*b*CosC
CosC=(a^2+b^2-c^2)/2ab
CosB=(a^2+c^2-b^2)/2ac
CosA=(c^2+b^2-a^2)/2bc
Prove:
∵ As shown in the figure, there is a→+b→=c→
∴c c=(a+b) (a+b)
∴c^2=a a+2a b+bb∴c^2=a^2+b^2+2|a||b|cos(π-θ)
Get c 2 = a 2+b 2-2 | a || b | cos θ (note: the formula of trigonometric function is used here).
Then disassemble it to get C 2 = A 2+B 2-2 * A * B * COSC.
The same principle can prove to others that the following CosC = (c 2-b 2-a 2)/2ab means to move CosC to the left.
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Proof method of plane geometry:
In any delta △ABC
Do AD⊥BC.
The opposite side of ∠C is C, the opposite side of B is B, and the opposite side of A is A.
There are BD=cosB*c, AD=sinB*c, DC = BC-BD = A-COSB * C.
According to Pythagorean theorem:
AC^2=AD^2+DC^2
b^2=(sinB*c)^2+(a-cosB*c)^2
b^2=sin^2b*c^2+a^2+cos^2b*c^2-2ac*cosb
b^2=(sin^2b+cos^2b)*c^2-2ac*cosb+a^2
b^2=c^2+a^2-2ac*cosB
cosB=(c^2+a^2-b^2)/2ac
From the cosine theorem and the properties of cosine function, we can see that if the sum of squares of two sides of a triangle is equal to the square of the third side, then the angle subtended by the third side must be a right angle; If it is less than the square of the third side, the angle subtended by the third side is obtuse; If it is greater than the third side, the angle subtended by the third side is acute. That is, the shape of triangle can be judged by cosine theorem. At the same time, we can also use cosine theorem to find the range of triangle side length.
pythagorean theorem
Pythagorean theorem:
In China, the property that the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse is called Pythagorean theorem or Pythagorean theorem. Is a basic geometric theorem, which is traditionally considered to be proved by Pythagoras in ancient Greece. It is said that after Pythagoras proved this theorem, he beheaded a hundred cows to celebrate, so it is also called "Hundred Cows Theorem". In China, Zhou Kuai Shu Jing recorded a special case of Pythagorean Theorem, which was reportedly discovered by Shang Dynasty's Shang Dynasty, so it is also called Shang Gao Theorem. Zhao Shuang in the Three Kingdoms period made detailed comments on Pythagorean Theorem as proof in Zhou Bi suan Jing. France and Belgium are called donkey bridge theorem, and Egypt is called Egyptian triangle. In ancient China, the shorter right angle side of a right triangle was called a hook, the longer right angle side was called a chord, and the hypotenuse was called a chord.
Theorem:
If the two right angles of a right triangle are A and B and the hypotenuse is C, then A+B = C; That is to say, the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.
If the three sides A, B and C of a triangle satisfy a+b=c, then the triangle is a right triangle. (called the inverse theorem of Pythagorean theorem)
The earliest Pythagorean theorem
According to many clay tablets, Babylonians were the first people in the world to discover Pythagoras theorem. This is just an example. For example, in BC 1700, the ninth question on a clay tablet (No.BM85 196) was to the effect that "there is a wooden beam (AB) with a length of 5 meters vertically leaning against the wall, and the upper end (A) slides down one meter to d .. How far is the lower end (C) from the wall root (B)?" They solved the problem with Pythagorean theorem, as shown in the figure.
Let AB = CD = L = 5m, BC=a, AD = H = 1m, BD = L-H = 5- 1m = 4m.
A = √ [L-(L-H)] = √ [5-(5-1)] = 3m, ∴ Triangle BDC is a twisted shape with three sides, four sides and five sides.
Brief Introduction of Zhou Kuai suan Jing
The Pythagorean Zhou Kuaijing is one of the ten books of calculation. Written in the second century BC, it was originally named Zhou Jie, which is the oldest astronomical work in China. It mainly expounded the theory of covering the sky and the method of four seasons calendar at that time. In the early Tang Dynasty, it was stipulated as one of imperial academy's teaching materials, so it was renamed Zhou Kuai. The main achievement of Zhouyi ·suan Jing in mathematics is the introduction of Pythagorean theorem and its application in measurement. The original book did not prove Pythagorean theorem, but the proof was given by Zhao Shuang in Zhou Zhuan Pythagorean Notes. ·suan Jing of Zhouyi adopts quite complicated fractional algorithm and Kaiping method.
[Edit this paragraph] Garfield's story of proving Pythagorean theorem
1876 One weekend evening, on the outskirts of Washington, D.C., a middle-aged man was walking and enjoying the beautiful scenery in the evening. He was Ohio and party member Garfield. Walking, he suddenly found two children talking about something with rapt attention on a small stone bench nearby, arguing loudly and discussing in a low voice. Driven by curiosity, Garfield followed the sound and came to the two children to find out what they were doing. I saw a little boy bend down and draw a right triangle on the ground with branches. So Garfield asked them what they were doing. The little boy said without looking up, "Excuse me, sir, if the two right angles of a right triangle are 3 and 4 respectively, what is the length of the hypotenuse?" Garfield replied, "It's five." The little boy asked again, "If the two right angles are 5 and 7 respectively, what is the length of the hypotenuse of this right triangle?" Garfield replied without thinking, "The square of the hypotenuse must be equal to the square of 5 plus the square of 7." The little boy added, "Sir, can you tell the truth?" Garfield was speechless, unable to explain, and very unhappy.
So Garfield stopped walking and immediately went home to discuss the questions the little boy gave him. After repeated thinking and calculation, he finally figured it out and gave a concise proof method.
As follows:
Solution: In the grid, the area of two small triangles with right angles is equal to the area of a triangle with hypotenuse.
The content of Pythagorean theorem: the sum of squares of two right angles A and B of a right triangle is equal to the square of hypotenuse C,
a^2； +b^2； =c^2；
Explanation: Ancient scholars in China called the shorter right-angled side of right-angled triangle "hook", the longer right-angled side "chord" and the hypotenuse "chord", so they called this theorem "Pythagorean Theorem". Pythagorean theorem reveals the relationship between the sides of a right triangle.
For example, if the two right angles of a right triangle are 3 and 4 respectively, then the hypotenuse c= a+b=9+ 16=25.
Then the hypotenuse is 5.
Some exercises of Pythagorean theorem
Chapter 1 Pythagorean Theorem 1. The content of Pythagorean Theorem, how did Pythagorean Theorem get, and what inspiration did you get from the proof of the Theorem?
Exercise:
1. In △ABC, ∞∠C = 90. (1+0) If A = 2 and B = 3, what is the area of a square with C as its side? (2) If A = 5 and C = 13, what is B? (3) If c = 6 1 and b = 1 1, what is a? (4) If a∶c =3∶5 and c =20, what is B? (5) If ∠ A = 60 and AC =7cm, AB = _cm and BC = _cm.
2. A right side and hypotenuse of a right triangle are 8cm and 10cm respectively, so the height on the hypotenuse is higher than _ cm.
3. The circumference of an isosceles triangle is 20cm, the base height is 6cm and the base length is _ cm.
4. In △ABC, if AD = _cm, ∠ BAC = 120, AB = 12 cm, the height of BC is _cm.
5. When △ABC, ∠ ACB = 90, CD⊥AB is in D, BC=, DB=2cm, then BC = _ BC=_ cm, AB= _cm, AC= _cm _ cm.
6. As the picture shows, someone has thought about crossing the river. Due to the influence of ocean current, the actual landing point C deviates from the scheduled arrival point B200m. As a result, he actually swam 520m in the water, and the width of the river was _ _ _ _ _.
7. There are two monkeys at the height of 10 meter of a tree. A monkey climbed down the tree and walked to the pond 20 meters away from the tree. The other climbs to the top of the tree D and jumps directly to A, and the distance is calculated in a straight line. If two monkeys pass the same distance, the tree is _ _ _ _ _ _ _ meters high.
8. Given an Rt△, the two sides are 3 and 4 respectively, and the square of the third side is ().
A, 25 B, 14 C, 7 D, 7 or 25
Xiaofeng's mother bought a 29-inch (74 cm) TV set. Which of the following statements about 29 inches is correct?
A. Xiaofeng thinks it refers to the length of the screen; B. Xiaofeng's mother thinks it refers to the width of the screen;
C. Xiaofeng's father thinks it refers to the circumference of the screen; D. Salesman thinks it refers to the diagonal length of the screen.
2. How many ways do you prove that a triangle is a right triangle?
Exercise:
(× classic exercises ×)
According to China's ancient "Zhou Kuai Shu Jing", Shang Gao told the Duke of Zhou in 1 120 BC that if a ruler is folded into a right angle, the two ends are connected to form a right triangle. If the hook is three and the strand is four, then the string is equal to five, which is summarized as "hook three, strand four and string five" by later generations.
(1) Observation: 3, 4, 5, 5, 12, 13, 7, 24, 25, ... It is found that the ticks of these groups are all odd, and they have not stopped since 3. Calculate 0.5 (9+ 1) and 0.5 (25- 1) and 0.5 (25+ 1), and write the formulas of strands and chords that can represent the three numbers of 7, 24 and 25 respectively according to the rules you found.
(2) According to (1) law, if all hooks of Pythagorean number are represented by n(n is odd, n≥3), please directly represent their chords by algebraic expressions containing n..
Answer:
( 1) 0.5(9+ 1)∧2+0.5(25- 1)∧2= 169=0.5(25+ 1)∧2 0.5( 13+ 1)∧2+0.5(49- 1)∧2=0.5(49+ 1)∧2
(2) Chord: 0.5 (n 2- 1) Chord: 0.5 (n 2+ 1)
If the three sides of a triangle are (a+b)2=c2+2ab, then the triangle is ().
A. equilateral triangle; B. obtuse triangle; C. right triangle; D. acute triangle
1. in Δ Δ ABC, if AB2+BC2 = AC2, then ∠ A+∠ C = 0.
2. As shown in the figure, if the side length of a small square is 1, the △ABC in the square grid is ().
(a) right triangle (b) acute triangle
(c) obtuse triangle (d) None of the above answers are correct
Given that the lengths of three sides of a triangle are 2n+ 1, 2n+2n, 2n+2n+ 1 (n is a positive integer), the maximum angle is equal to _ _ _ _ _ _ _.
The degree ratio of the three internal angles of a triangle is 1:2:3, and its largest side is m, so its smallest side is _ _ _ _.
The area of an isosceles right triangle with the hypotenuse height of m is equal to _ _ _ _.
3. As shown in the figure, in quadrilateral ABCD, AB=3cm, AD=4cm, BC= 13cm, CD= 12cm, ∠ A = 90, find the area of quadrilateral ABCD.
There is a very important theorem in trigonometry, which is called Pythagorean Theorem and Quotient Theorem in China. Because it is mentioned in "Weekly Parallel Calculations", Shang Gao said "hook three strands, four strings and five". Here are some proofs.
It turns out that there are differences. Let A and B be the right sides of a right triangle, and C is the hypotenuse. Consider the squares A and B with a+b on both sides in the picture below. Divide A into six parts and B into five parts. Since eight small right-angled triangles are congruent, it can be deduced that the square of the hypotenuse is equal to the sum of the squares of two right-angled sides by subtracting the equivalence from the equivalence. Here, the quadrilateral in B is a square with a side length of c, because the sum of the three internal angles of a right triangle is equal to two right angles. The above proof method is called subtraction congruence proof method. Chart B is a "chord chart" in the classic weekly parallel computing.
The following figure is the proof given by H. Perigal in 1873, which is an additive congruence proof method. In fact, this proof was rediscovered, because Labitibn Qorra (826 ~ 90 1) already knew this division. (such as the picture on the right) One of the following proofs is H? 6? 1E？ 6? 1 was given by Du Deni on 19 17. It is also a proof method of adding congruence.
As shown in the picture on the right, the square area with side length b plus the square area with side length a equals the square area with side length c.
The proof method in the picture below is said to be L? 6? 1? 6? Designed by1Vinci (1452 ~1519), the proof method of subtraction congruence is adopted.
Euclid gave an extremely ingenious proof of Pythagorean theorem in proposition 47 of Volume I of Elements of Geometry, such as the picture on the next page. Because of the beautiful graphics, some people call it "the friar's headscarf" and others call it "the bride's sedan chair", which is really interesting. Professor Hua once suggested sending this photo into the universe to communicate with "aliens". The outline of the certificate is:
(AC)2=2△JAB=2△CAD=ADKL .
Similarly, (BC)2=KEBL
therefore
(AC)2+(BC)2=ADKL+KEBL=(BC)2
Indian mathematician and astronomer Bascara (active around 1 150) gave a wonderful proof of Pythagorean theorem and a split proof. As shown in the figure below, divide the square on the hypotenuse into five parts. Four of them are triangles that are congruent with a given right triangle; One part is a small square with the difference between two right-angled sides as the side length. It is easy to put these five parts together again and get the sum of the squares of two right angles. In fact,
Poshgaro also gives a proof of the following figure. Draw the height on the hypotenuse of a right triangle to get two pairs of similar triangles, and you have it.
c/b=b/m，
c/a=a/n，
cm=b2
cn=a2
Add up both sides.
a2+b2=c(m+n)=c2
This proof was rediscovered in the17th century by the British mathematician J. Wallis (Wallis,161703).
Several American presidents have a subtle connection with mathematics. g？ 6? Washington was once a famous surveyor. t？ 6? Jefferson vigorously promoted American higher mathematics education. Lincoln studied logic by studying Euclid's Elements of Geometry. Even more creative is J.A. Garfield (183 1 ~ 1888), the 7th president. He had a strong interest in elementary mathematics and superb talent when he was a student. 1876, (then a congressman, elected president of the United States five years later) gave a beautiful proof of Pythagorean theorem, which was published in the New England Journal of Education. The idea of proof is to use the area formulas of trapezoid and right triangle. As shown on the next page, it is a right-angled trapezoid composed of three right-angled triangles. Find the same area with different formulas
that is
a2+2ab+b2=2ab+c2
a2+b2=c2
This kind of proof is often of interest to middle school students when they study geometry.
There are many ingenious proofs of this theorem (it is said that there are nearly 400 kinds). Here are some examples for students, all of which are proved by puzzles.
The certificate 1 is shown in Figure 26-2. On the outside of the right triangle ABC, make squares ABDE, ACFG and BCHK, and their areas are c2, b2 and a2 respectively. We just need to prove that the area of a big square is equal to the sum of the areas of two small squares.
Draw CM‖BD through C, cross AB to L, and connect BC and CE. because
AB=AE，AC=AG ∠CAE=∠BAG，
So △ ace△ agb
SAEML=SACFG ( 1)
The same method can also be proved.
SBLMD=SBKHC (2)
( 1)+(2)
SABDE=SACFG+SBKHC，
Namely c2=a2+b2
Proof 2 is shown in Figure 26-3 (Figure Zhao). Eight right-angled triangles ABC are used to form a big square CFGH with a side length of a+b, and there is an inscribed square ABED with a side length of C, as shown in the figure.
SCFGH=SABED+4×SABC，
So a2+b2=c2
Proof 3 is shown in Figure 26-4 (map of Mei Wending).
Draw a square ABDE outward on the hypotenuse AB of the right angle △ABC and a square ACGF on the right angle AC. It can be proved that (slightly) expanding GF must pass E; Extend CG to K, make GK=BC=a, connect KD, and make DH⊥CF in H, then DHCK is a square with a side length of A.
The area of a Pentagon
On the one hand,
S= square ABDE area +2 times △ABC area.
=c2+ab ( 1)
On the other hand,
S= square ACGF area+square DHGK area
+2 times △ABC area
=b2+a2+ab。 (2)
Derived from (1) and (2)
c2=a2+b2
Proof 4: As shown in Figure 26-5 (Mingda Xiang diagram), a square ABDE is made on the hypotenuse of right triangle ABC, and a square BFGJ with side length b is completed on the basis of two right angles CA and CB of right triangle ABC (Figure 26-5). It can be proved that the extension line of (omitted) GF must pass through D. Extend AG to K, make GK=a, and let EH⊥GF be H, then EKGH must be a square with a side length equal to A.
Let the area of pentagonal EKJBD be S. On the one hand,
S=SABDE+2SABC=c2+ab ( 1)
On the other hand,
S=SBEFG+2？ 6? 1S△ABC+SGHFK
=b2+ab+a2
Pass (1), (2)
Lead to an argument
They are all verified by area: a large area is equal to the sum of several small areas. The equation is obtained by different representations of the same area, and then the Pythagorean theorem is simplified. See/21010000/VCM/0720gdl. Doctor.
Pythagorean theorem is one of the most proven theorems in mathematics-there are more than 400 proofs! But the first recorded proof-Pythagoras's proof method has been lost. The earliest proof that can be seen at present belongs to the ancient Greek mathematician Euclid. His proof is in the form of deductive reasoning, which is recorded in the mathematical masterpiece "Elements of Geometry". Among the mathematicians in ancient China, Zhao Shuang, a mathematician from the State of Wu in the Three Kingdoms period, was the first to prove the Pythagorean theorem. Zhao Shuang created Pythagorean Square Diagram, and gave a detailed proof of Pythagorean Theorem by combining numbers and shapes. In this Pythagorean Square Diagram, the square ABDE with the chord as the side length is composed of four equal right triangles and a small square in the middle. The area of each right triangle is AB/2; If the side length of the small square in the middle is b-a, the area is (b-a) 2. Then we can get the following formula: 4×(ab/2)+(b-a) 2 =c 2. After simplification, we can get: a 2 +b 2 =c 2, that is, c=(a 2 +b 2) (1/2). Zhao Shuang's proof is unique and innovative. He proved the identity relationship between algebraic expressions by cutting, cutting, spelling and supplementing geometric figures, which was rigorous and intuitive, and set a model for China's unique ancient style of proving numbers by shape, unifying numbers by shape, and closely combining algebra and geometry. The following website is Zhao Shuang's pythagorean chart:/catchpic/0/01f9d756be31E31e71a75cacc14/kloc-0. Most mathematicians after GIF have inherited this. For example, Liu Hui later proved Pythagorean theorem by formal proof. Liu Hui used the "in-out complementary method", that is, the cut-and-paste proof method. He cut out some areas on the square with Pythagoras as the edge and moved them to the blank areas with chords as the edge. The result was just filled in, and the problem was completely solved by graphic method. The following website is Liu Hui's Green Zhu Access Map:/catchpic/a/A7/A7070D771214459D67A 75E8675A A4DCB.gif.
Pythagorean theorem is widely used. Another ancient book in the Warring States period in China, Twelve Notes on the Postscript of the History of the Road, recorded this: "Yu ruled the flood and decided to flow in the rivers, watching the shape of mountains and rivers, and decided to compete. In addition to catastrophic disasters, the East China Sea is flooded and there is no danger of drowning. " In order to control the flood, Dayu decided the direction of the water flow according to the height of the terrain, guided the situation to make the flood flow into the sea, so that there would be no more flood disaster. This is the result of applying Pythagorean theorem.
Pythagorean theorem is widely used in our life.
Verification method of 16 Pythagorean theorem (with pictures):/uploadfiles/2007/11-25/1125862269. document
Exercise: an isosceles triangle, the ratio of three internal angles is1:1:kloc-0/0, and the waist length is 10cm, then the area of this triangle is _ _ _ _.
Solution: The angles of the triangle are 15 degrees and 150 degrees respectively.
Let the height of the bottom edge be h and the length of the bottom edge be 2t.
It is easy to get sin15 = sin60cos45-cos60sin45 = h/10.
The solution is h=5(√6-√2)/2.
tan 15 =(tan 60-tan 45)/( 1-tan 60 tan 45)= 5(√6-√2)/2t。
T=5(√6+√2)
Therefore, the area s = th = 50.
Alias of Pythagorean Theorem
Pythagorean theorem is a dazzling pearl in geometry, which is called "the cornerstone of geometry" and is also widely used in higher mathematics and other disciplines. Because of this, several ancient civilizations in the world have been discovered and widely studied, so there are many names.
China is the first country to discover and study Pythagorean theorem. Ancient mathematicians in China called right triangle pythagorean, the shorter right side called hook, the other right side called chord, and the hypotenuse called chord, so pythagorean theorem is also called pythagorean chord theorem. In BC 1000 years, according to records, Shang Gao (about BC 1 120 years) answered the Duke of Zhou. In the right triangle, there are "three hooks, four strands and five strings". Therefore, Pythagorean theorem is also called "quotient height theorem" in China. In the 7th-6th century BC, China scholar Chen Zi once gave the trilateral relationship of any right triangle, that is, "The sun is the hook, the sun is the share, and the hook and the share are multiplied and divided, and the evil belongs to the sun.
In France and Belgium, Pythagorean Theorem is also called Donkey Bridge Theorem. Other countries call Pythagorean Theorem Square Theorem.
One hundred and twenty years after Chen Zi's death, the famous Greek mathematician Pythagoras discovered this theorem, so many countries in the world called it Pythagoras theorem. In order to celebrate the discovery of this theorem, the Pythagorean school killed one hundred cows as a reward for offering sacrifices to the gods, so this theorem is called "Hundred Cows Theorem".