pythagorean theorem

Understand the proof of Pythagorean Theorem, master the content of Pythagorean Theorem, and preliminarily use Pythagorean Theorem for relevant calculation, drawing and proof.

2. Through the application of Pythagorean theorem, we can cultivate the thinking and logical reasoning ability of equations.

3. Compare and introduce the research of Pythagorean Theorem between ancient mathematicians in China and western mathematicians, and educate students in patriotism.

Teaching emphases and difficulties

The emphasis is on the application of Pythagorean theorem; The difficulty is the proof and application of Pythagorean theorem.

Teaching process design

First, stimulate the interest in introducing topics.

This paper introduces China mathematician Hua's suggestion of sending Pythagorean Theorem to the universe to communicate with aliens, and shows that Pythagorean Theorem was discovered by ancient mathematicians in China 2000 years ago, which aroused students' interest and pride in Pythagorean Theorem and introduced the subject.

Second, the exploration, proof and naming of Pythagorean theorem

1. Guess the conclusion.

What is the content of Pythagorean theorem? Please also experience the joy of mathematicians discovering new knowledge.

Teachers use computers to demonstrate:

(1) In △ABC, the opposite sides of ∠A, ∠B and ∠C are A, B and C respectively, and ∠ ACB = 90, which makes △ABC move, but always keeps ∠ ACB = 90, such as dragging A.

(2) In the above process, always measure a2, b2 and c2, and list the measured values (about 7 ~ 8) of one or two states of the above typical actions in a table, so that students can observe the quantitative relationship among the three numbers and get a guess.

(3) The comparison shows that there is no such relationship between the squares of three sides of acute triangle and obtuse triangle, so it is a unique property of right triangle. Let the students describe his guess in words, draw a picture, write what is known and verify it.

2. Prove the conjecture.

There are hundreds of ways to prove Pythagorean theorem in the world. Even Garfield, the 20th president of the United States, provided the area proof method in 188 1 (see figure (4) on page 109 of the textbook), while the ancient mathematicians in China used the idea of graphic cutting and splicing to calculate the area and provided many proof methods. We use one of them (the teacher makes a demonstration of teaching AIDS).

3. Naming Pythagorean Theorem.

China called this conclusion Pythagorean Theorem, and the West called it Pythagorean Theorem. Why?

(1) introduces the record of Pythagorean theorem in Zhouyi Shu Jing;

(2) Pythagoras discovered the Pythagorean theorem in 582-493 BC.

(3) Compare the above facts, educate students in patriotism and encourage them to make progress.

Third, the application of Pythagorean theorem

1. It is known that any two sides of a right triangle can find the third side.

Example 1 in Rt△ABC, the opposite sides of ∠C = 90, ∠A, ∠B and ∠C are a, b and c respectively.

(1) A = 6, B = 8 Find the height of C and the hypotenuse; (2) a = 40, c = 4 1, and find b; (3) b = 15, = 25 to find a; (4) A: B = 3: 4, C = 15, and find B. 。

Note: For (1), students are required to summarize the basic method of calculating the height of the hypotenuse by using the area in the basic graph (Figure 3- 153); For (4), guide students to use the idea of equations to solve problems.

The teacher writes (1) and (4) on the blackboard and asks the students to practice (2) and (3).

Example 2 Find the distance (accurate to 0. Lmm) is between the centers A and B of two holes in the rectangular part shown in Figure 3- 152 (in mm).

The teacher showed how to find the projection of the known conditions in the right triangle ABC according to the size on the diagram.

Exercise 1 projection shows: (1) at the isosceles Rt△ABC, ∠ C = 90, AC: BC: AB = _ _ _ _ _ _ _;

(2) As shown in Figure 3- 153 ∠ ACB = 90 and ∠ A = 30, then BC: AC: AB = _ _ _ _ _ _ _ _; If AB = 8, then AC = _ _ _ _ _ _ _ _ _ _ _ _ _； If CD⊥AB, CD = _ _ _ _ _ _ _ _ _ _ _ _

(3) If the side length of equilateral △ABC is a, the height AD = _ _ _ _ _ _ _ _ _ _ _ _

S △ABC=______________

Description:

(1) Learn to solve problems with the idea of equations.

(2) Through this question, let students summarize and be familiar with some common conclusions in basic graphics:

① The ratio of three sides of an isosceles right triangle is1:1:;

② The ratio of three sides of a right triangle with an angle of 30 is1::2;

③ The height of an equilateral triangle with a side length of A is A, and the area is

Example 3 (blackboard writing) is shown in Figure 3- 154, AB = AC = 20, BC = 32, and△ DAC = 90. Find the length of BD.

Analysis:

(1) decomposes the basic graph, which includes isosceles △ABC and.

rt△ADC；

(2) Add an auxiliary line-the height of the bottom of the isosceles delta △△ABC.

AE, which is also the height on the hypotenuse of Rt△ADC;

(3) Let BD be X. Using the basic relationship in Figure 3- 153,

Solve by column equation. The detailed process of the teacher writing on the blackboard.

The solution is AE⊥BC in E. Let BD be X, then DE= 16-x, AE2=AC2-EC2, AD2=DE2+AE2=DC2-AC2. Substituting the above formula, DE2+AC2-EC2=DC2-AC2, that is, 2AC2 =

∴2×202 =(32-x)2+ 162-( 16-x)2，x=7。

2. Drawing with Pythagorean Theorem.

Example 4 is a section with a length of.

Note: Just analyze and draw according to the textbook page 10 1, and emphasize the construction method of right triangle and define the unit length by yourself.

3. Prove by Pythagorean theorem.

Example 5 is shown in figure 3- 155, in △ABC, CD⊥AB is in D, AC >；; BC.

Verification: ac2-bc2 = ad2-bd2 = ab (ad-BD).

Analysis:

(1) The Pythagorean theorem is used to decompose the right triangle.

In Rt△ACD, AC2 = Ad2+Cd2；; In Rt△BCD, BC2 = Cd2+BD2.

(2) sorting by using the constant deformation technique in algebra:

AC2-BC2=(AD2+CD2)-(CD2+BD2)

=AD2-BD2

=(AD+BD)(AD-BD)

=AB(AD-BD)。

Example 6 is known: as shown in figure 3- 156, RT △ ABC, ∠ ACB = 90, d is the midpoint of BC, and DE⊥AB is in E. Verification: AC2 = AE2-BE2.

Analysis: Add auxiliary connection AD, construct two new right triangles, and choose Pythagorean theorem and expressions related to the conclusion to prove it.

4. Optional examples.

(1) As shown in Figure 3- 157, in Rt△ABC, ∠ C = 90, ∠ A = 15, BC = 1. Find the area of △ABC.

Tip: Add an auxiliary line-the perpendicular of BA intersects with Ba at D, AC intersects with E, and connect BE to form a right triangle BCE with an included angle of 30, and solve it by Pythagorean theorem at the same time, or directly make ∠ Abe = 15 in ∠ABC and intersect with CA at E. 。

(2) as shown in figure 3- 158, △ABC, ∠ A = 45, ∠ B = 30, BC = 8. Find the length of the AC side.

Analysis: Add auxiliary lines-CD⊥AB in D, and construct right triangle equations with 45 and 30 angles to solve.

(3) As shown in Figure 3- 159(a), in quadrilateral ABCD, ∠B=

∠ d = 90, ∠ c = 60, AD= 1, BC=2, find AB, CD.

Tip: Add auxiliary lines-extend the intersection of BA and CD at point E, and construct RT △ EAD and RT △ EBC with 30 angle. Use their characteristics to solve the problem (see Figure 3- 159(b)). Or divide the quadrilateral ABCD into a right-angled triad and a 30-degree rectangle to solve the problem. (See Figure 3-65438)

Answer: AB=23-2, CD = 4-3.

(4) Known: 3- 160(a), rectangular ABCD .. (all four corners are right angles)

①P is a point in a rectangle, which proves that PA2+ PC2= PB2+ PD2.

② Whether the conclusion still holds when P moves to the edge of AD (Figure 3- 160 (b)) and outside the rectangular ABCD (Figure 3- 160 (c)).

Analysis:

(1) Add auxiliary lines to each of the four right-angled triangles-passing through P means EF⊥BC, and passing through BC means f.

Using Pythagorean Theorem.

(2) The three questions can be summarized into one proposition as follows:

The sum of squares of distances from any point to non-adjacent vertices on a rectangular plane is equal.

Fourth, the summary of teachers and students' common memories.

The content and proof method of 1. Pythagorean theorem.

2. The function of Pythagorean Theorem: The shape characteristics of a triangle (one angle is 90) can be transformed into a quantitative relationship, that is, three sides satisfy a2+b2=c2.

3. When using Pythagorean theorem to calculate and prove, we should pay attention to finding the relevant line segments of right triangle with the idea of equation.

Dragon; Using auxiliary lines to construct right triangle, using Pythagorean theorem.

Verb (short for verb) homework

1. Textbook page 106, questions 2-8.

2. Look at the textbook page 109: proof of Pythagorean theorem.

Description of classroom teaching design

This instructional design takes 2 class hours to complete.

1. Pythagorean theorem reveals the quantitative relationship among three sides of right triangle, which is an important property of right triangle. This teaching design makes use of the superior conditions of the computer (dynamically displayed by the geometric sketchpad software) to provide enough typical materials-right triangles with various shapes, sizes and positions, so that students can observe and analyze, summarize, explore the relationship among the three sides of right triangles and compare them with acute and obtuse triangles.

2. Schools can also introduce new courses by adopting the following exploration methods of analogy and association according to their own teaching conditions.

(1) Review the relationship between the three sides of a triangle and summarize the rule that the sum of the smaller two sides is greater than the third side.

(2) Guide students to analogize: What is the relationship between the sum of squares of the smaller two sides and the square of the third side?

(3) Give three examples (see Figure 3- 16 1 (a) (b) (c)).

By comparison, it is found that the sum of squares of two smaller sides in acute triangle and obtuse triangle is greater than or less than the square of the third side, and the sum of squares of two smaller sides in right triangle is equal to the square of the third side.

(4) Demonstrate Figure 3- 15 1 with teaching AIDS to verify the right triangle conjecture.

Teaching purpose: 1, will explain the inverse theorem of Pythagorean theorem.

2. The inverse theorem of Pythagorean theorem will be used to determine whether a triangle is a right triangle.

3. Be able to apply Pythagorean theorem and Pythagorean inverse theorem correctly and flexibly.

Teaching Emphasis: Application of the Inverse Theorem of Pythagorean Theorem

Teaching Difficulties: Proof of the Inverse Theorem of Pythagorean Theorem

Teaching method: combining lectures with practice.

Teaching process:

First, review questions

1, the written language of Pythagorean theorem

2. Geometric symbolic language of Pythagorean theorem

3. The function of Pythagorean theorem

4. Fill in the blanks: both sides of a given right triangle are 5, 12, and the length of the third side is.

Second, the introduction of new courses.

Pythagorean theorem is a proposition, and any proposition has an inverse proposition. What is its inverse proposition?

Third, explain the new lesson.

Written language of Pythagorean Theorem Inverse Theorem: If the lengths of three sides of a triangle: A, B and C are related, and a2+b2=c2, then the triangle is a right triangle.

A proposition can be divided into true proposition and false proposition. We have to prove whether it is true or not.

It is known that in ABC, AB=c, BC=a, CA=b, a2+b2=c2.

Verification: ∠C=90?

Analysis: Prove that an angle is 90? , which can prove that AC⊥BC

You can also prove it with books and teach yourself.

It is proved that the inverse proposition of Pythagorean theorem is true, that is, the inverse theorem of Pythagorean theorem.

Geometric symbolic language of the inverse theorem of Pythagorean theorem: in Δ ABC ∶ A2+B2 = C2 (or C2-A2 = B2)

∴∠C=90？ (Inverse Theorem of Pythagorean Theorem)

Key point: as long as the above relationship is satisfied, it must be a right triangle, the long side is the hypotenuse, and the angle it faces is the right angle.

For example, if three sides are 3, 4 and 5, can they form a right triangle, 5, 12, 13? 9, 40, 4 1?

Pythagorean number: it can be three positive integers with the length of three sides of a right triangle, which is called Pythagorean number (or Pythagorean chord number).

Book 102- 103, draw the definition, and finish your homework 103, page 1, page 3.

For example, the three sides of1Δ ABC are the following groups of values, which can form a right triangle and indicate which one is right, otherwise it is marked with ×.

⑴a= 1、b=、c= 1

⑵a= 1.2、b= 1.6、c=2

⑶a:b:c=2: :2

⑷a=n2- 1、b=2n、c= n2+ 1(n> 1)

5] a = 2n2+ 1, b=2n2+2n, c = 2mn (m > n) m and n are positive integers.

The solution (1) ∫12+12 = () 2 ∴ΔABC is a triangle with ∠B as the right angle.

⑵ ∵22- 1.62=(2+ 1.6)(2- 1.6)= 1.44=( 1.2)2

∴δABC is a triangle with∠Δb as the right angle.

(3) (4) (5) Interpretation.

Emphasis: for large numbers, simple operations can be realized by square difference formula.

Example 2 is known: As shown in the figure, AD=3, AB=4, ∠BAD=90? ，BC= 12，CD= 13，

Find the area of quadrilateral ABCD.

Analysis: connect BD and find BD=5.

∵BD2+BC2=CD2 ∴∠CBD=90？

∴ area of quadrilateral ABCD = area of δ Abd+area of δ BD.

Solution: Omit

Example 2 is known: As shown in the figure, in ABC, CD is the height on the side of AB, and CD2=AD2? Bachelor of science

Prove that ABC is a right triangle.

Analysis: Prove that ABC is a right triangle.

Just prove that AC2+BC2=AB2.

In rt δ ACD, ∫∠ACD = 90?

∴AC2=AD2+CD2

Similarly, BC2=CD2+BD2.

∴AC2 + BC2 = AD2+2 CD2+BD2

=(AD+BD)2

∴δabc is a right triangle.

Let the students complete the proof process by themselves.