Knowledge points and typical examples of Pythagorean theorem in the second volume of the eighth grade of People's Education Press
First, the basic knowledge points:
1. Pythagorean theorem
Content: The sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse;
Representation: If two right angles of a right triangle are and the hypotenuse is, then
The origin of Pythagorean theorem: Pythagorean theorem, also known as quotient height theorem, is called Pythagorean theorem in the west. In ancient China, the shorter right-angled side of a right-angled triangle was called a hook, the longer right-angled side was called a chord, and the hypotenuse was called a chord. As early as 3000 years ago, Shang Gao, a mathematician in Zhou Dynasty, put forward the Pythagorean theorem in the form of "hook three, string four and string five". Later, people further discovered and proved that the relationship among the three sides of a right triangle is the sum of two right angles.
2. Proof of Pythagorean Theorem
There are many ways to prove Pythagorean theorem, and the common one is jigsaw puzzle.
The idea of verifying Pythagorean theorem with jigsaw puzzle method is as follows
(1) After cutting and mending, the area will not change as long as there is no overlap and no gap.
② According to the different representations of the same graphic area, the equations are listed and the Pythagorean theorem is deduced.
Common methods are as follows:
Method 1: Simplify the proof.
Method 2:
The sum of the areas of four right triangles and a small square is equal to the area of a big square. The sum of the area of four right-angled triangles and the area of a small square is the area of a big square, so method 3: simplify the proof.
3. The application scope of Pythagorean theorem
Pythagorean theorem reveals the quantitative relationship between three sides of right triangle, which is only applicable to right triangle, but not to three sides of acute triangle and obtuse triangle. Therefore, when applying Pythagorean Theorem, it must be clear that the object under investigation is a right triangle.
4. Application of Pythagorean Theorem ① Knowing the length of any two sides of a right triangle and finding the third side in the middle, then ② Knowing one side of a right triangle, we can get the quantitative relationship between the other two sides ③ Using Pythagorean Theorem to solve some practical problems.
5. Inverse theorem of Pythagorean theorem
If all three sides of a triangle are long enough, then the triangle is a right triangle, and the hypotenuse is the hypotenuse.
① The inverse theorem of Pythagorean theorem is an important method to judge whether a triangle is a right triangle. It determines the possible shapes of triangles by "transforming numbers into shapes". When applying this theorem, the sum of squares of two small sides can be compared with the square of the long side. If they are equal, then a triangle with three sides is a right triangle. If and when a triangle with three sides is an obtuse triangle; If, when, a triangle with three sides is an acute triangle;
(2) In the theorem, sum is only an expression and cannot be considered unique. If the three sides of a triangle are long enough, then a triangle with three sides is a right triangle, but it is a hypotenuse.
③ When the inverse theorem of Pythagorean Theorem is described by a problem, when the square of hypotenuse is equal to the sum of the squares of two right-angled sides, it cannot be said that this triangle is a right-angled triangle.
6. Pythagoras number
(1) Three positive integers that can form the three sides of a right triangle are called pythagorean numbers, that is, median numbers. When they are positive integers, they are called a set of Pythagorean numbers.
2 Remember that common Pythagorean numbers can improve the speed of solving problems, for example; ; ; wait for
(3) Representing the group pythagorean number by algebraic expression with letters;
(positive integer);
(is a positive integer) (,is a positive integer) 7. Application of Pythagorean Theorem
Pythagorean theorem can help us solve the problem of calculating the side length of right triangle or proving the relationship between line segments in right triangle. When using Pythagorean theorem, we must grasp the preconditions of a right triangle, know what the hypotenuse and right side of the right triangle are, and try to add auxiliary lines (usually vertical lines) to construct the right triangle, so as to correctly use Pythagorean theorem to solve it.
8. Application of the Inverse Theorem of Pythagorean Theorem
The inverse theorem of Pythagorean theorem can help us to judge whether a triangle is a right triangle by the quantitative relationship between its three sides. In the concrete calculation process, the sum of squares of two short sides should be compared with the sum of squares of the longest side, and the sum of squares of two sides should not be compared with the sum of squares of the third side without thinking, so as to draw a wrong conclusion.
9. Pythagorean theorem and the application of its inverse theorem
Pythagorean theorem and its inverse theorem are an inseparable whole when solving some practical problems or specific geometric problems. Usually, it is necessary to judge whether a triangle is a right triangle through the inverse theorem, and to calculate the side length by using the Pythagorean theorem. The two complement each other and complete the solution of the problem. Common graphics:
10, the concept of reciprocity proposition
If the topic and conclusion of one proposition are the conclusions and topics of another proposition, such two propositions are called reciprocal propositions. If one of them is called the original proposition, the other is called its inverse proposition.
Second, a classic example.
Question 1: directly investigate Pythagorean theorem
Example 1. Yes,.
(1) known length ..
⑵ Analysis of finding known solution length: directly apply Pythagorean theorem.
⑴
⑵
Question 2: Using Pythagorean Theorem to Measure Length
Example 1 If the bottom of the ladder is 9 meters away from the building, how many meters can a ladder with a length of 15 meters reach the building?
Analysis: This is a well-known problem of "knowing two to find one". After converting the physical model into the mathematical model, we can know the length of the hypotenuse and a right-angled side, and we can directly use Pythagorean theorem to find the length of the other right-angled side!
According to Pythagorean theorem AC2+BC2=AB2, namely AC2+92= 152, so AC2= 144, so AC= 12.
Example 2 is shown in Figure (8). In the pool, there is an upright reed at C, the distance from the shore D is 1.5m, and the length of the outlet BC is 0.5m. Pull the reed to the shore, and its top B just falls to the shore D, so as to find the depth AC of the pool.
Analysis: For example 1, first convert the physical model into a mathematical model, as shown in Figure 2. It can be seen from the meaning of the question that in △ACD, ∠ACD=90? SPAN & gtRt△ACD only knows CD= 1.5, which is a typical type of "knowing two and seeking one" by using Pythagorean theorem.
The standard problem solving steps are as follows (for reference only):
As shown in Figure 2, according to Pythagorean theorem, AC2+CD2=AD2.
Let the water depth AC= x meters, then AD=AB=AC+CB=x+0.5.
x2+ 1.52=( x+0.5)2
X=2 of the solution.
So the water depth is 2 meters.
Question 3: Pythagorean theorem and inverse theorem are used together-
Example 3 is shown in fig. 3. In a square ABCD, e is the midpoint of BC and f is a point on AB, so △DEF is a right triangle? Why?
Analysis: This question hides many conditions, which is a bit confusing at first glance. If you look at the problem carefully, you can find the law. We can create conditions without any conditions. If we can set AB=4a, then BE=CE=2 a, AF=3 a, BF= a, then the sum of Rt△AFD, Rt△BEF.
In Rt△CDE, the lengths of DF, EF and DE are calculated by Pythagorean theorem, and whether △DEF is a right triangle is judged by the inverse theorem of Pythagorean theorem.
Detailed problem solving steps are as follows:
Let the side length of square ABCD be 4a, then be = ce = 2a, af = 3a, BF = a.
In Rt△CDE, DE2=CD2+CE2=(4a)2+(2 a)2=20 a2.
Similarly, ef2 = 5a2 and df2 = 25a2.
In △DEF, EF2+ DE2=5a2+
20a2=25a2=DF2
△ def is a right triangle, and△ ∠ def = 90? SPAN & gt。
Note: this question uses the quadratic pythagorean theorem, which is a necessary exercise to master pythagorean theorem.
Question 4: Using Pythagorean Theorem to Find Line Length-
Example 4 is shown in fig. 4. It is known that AB = 8 cm and BC = 10 cm in rectangular ABCD. Take a point E on the edge CD, fold △ADE so that the point D just falls on the point F on the edge BC, and find the length of CE.
Analysis: Make clear the invariants in folding before solving the problem. Reasonable setting is the key.
The detailed problem solving process is as follows:
According to the meaning of the question, Rt△ADE≌Rt△AEF is obtained.
∴∠AFE=90？ SPAN & gt，AF= 10cm，EF=DE
Let CE=xcm,
Then de = ef = CD-ce = 8-X.
From Pythagorean Theorem of Rt△ABF;
AB2+BF2=AF2, that is, 82+BF2= 102.
∴BF=6cm
∴CF=BC-BF= 10-6=4(cm)
From Pythagorean Theorem of Rt△ECF;
EF2=CE2+CF2, that is, (8-x) 2 = x2+42.
∴64- 16x+x2=2+ 16
∴x=3(cm), that is, CE=3 cm.
Note: the length of the crease and the area of the overlapping part can also be found in this question.
Question 5: Use the inverse theorem of Pythagorean theorem to judge the vertical-
Example 5 As shown in Figure 5, Master Wang wants to check whether the AD side of the table is perpendicular to the AB side and the CD side. He measured AD = 80 cm, AB = 60 cm, BD = 100 cm. Is the AD side perpendicular to the AB side? How to verify whether the AD edge is perpendicular to the CD edge?
Analysis: Because the physical objects are generally large, it is not easy to measure the length with a ruler. We usually intercept a part of the length to verify. As shown in Figure 4, the rectangular ABCD represents the shape of the desktop, with AM= 12cm on AB and AN=9cm on AD (why do you want to set these two lengths? ), connect MN and measure the length of MN.
① If MN= 15, then AM2+AN2=MN2, so the AD side is perpendicular to the AB side;
② if MN=a≠ 15, then 92+122 = 81+144 = 225, A2 ≠ 225, that is, 92+122 ≠.
A2, so ∠A is not a right angle. Solving practical problems with Pythagorean theorem-
There is an induction-controlled lamp, which is installed on the wall above the door, 4.5 meters from the ground. As long as anything moves within 5 meters, the light will turn on automatically. How far does a student with a height of 1.5 meters have to walk to get to the door?
Analysis: First of all, it is necessary to find out whether the Chu people walked 5 meters from the lamp or 5 meters from the lamp. It is conceivable that the head should be 5 meters away from the lamp. It is converted into a mathematical model, as shown in Figure 6. Point A stands for control light, BM stands for human height, BC∥MN,BC⊥AN is 5 meters away from the head (point B), and find the length of BC. Ann is known.
Question 6: Rotation problem:
Example 1. As shown in the figure, △ABC is a right triangle and BC is the hypotenuse. After rotating △ABP counterclockwise around point A, it can coincide with △ACP'. If AP=3, find the length of PP'.
Variant 1: As shown in the figure, P is a point in the equilateral triangle ABC, PA=2, PB=, PC=4, and find the side length of △ABC.
Analysis: Using rotation transformation, choose 60 angstroms counterclockwise around point B as △BPA. What? Mechanical fermentation? Hey? Chievo school? /SPAN>。
According to their quantitative relationship, we can know that this is a right triangle from Pythagorean theorem.
Variant 2, as shown in the figure, △ABC is an isosceles right triangle, ∠BAC=90 angstrom? SPAN & gte and F are points on BC, and ∠EAF=45 angstroms? /SPAN>。
Try to discuss the relationship between them and explain the reasons.
Question 7: About folding.
For example 1, as shown in the figure, the side length of rectangular paper ABCD is AB = 10 cm, BC = 6 cm, and E is a point above BC. Fold the rectangular paper in half along AE, and the point B just falls on the point G on the edge of CD to find the length of BE.
Variant: As shown in the figure, AD is the center line of △ABC, ∠ADC=45 A? Choice? /SPAN>。 The ADC is folded along the straight line AD, and the point C falls at the position of the point C', BC=4. Find the length of BC.
Question 8: On the application of Pythagorean theorem in practice;
Example 1. As shown in the figure, MN Highway and PQ Highway meet at point P, and there is a middle school at point A, AP = 160m. The distance from point A to MN Expressway is 80m. If the tractor will be affected by noise within the range of 100m, then whether the school will be affected when the tractor runs along the PN direction on the MN expressway, please explain. If the speed of the tractor is known as 18km/h, how long will the school be affected?
Question 9: About the shortest question.
Example 5: As shown in the right figure 1- 19, the gecko found a pest at the bottom edge A of an oil tank with a bottom radius of 2m and a height of 4m, and decided to catch the pest. In order not to attract the attention of pests, it deliberately bypassed the oil tank and suddenly attacked pests from behind along the spiral route. (π takes 3. 14, and the result remains 1 bit after the decimal point, which can be calculated by a calculator. Variant: As shown in the figure, a cube with a side length of 3cm is divided into 9 small squares with a side length of 1cm. Suppose an ant crawls 2 cm per second, and it takes at least several hours to crawl from point A on the ground below to point B on the right.
Third, after-school training:
Fill in the blanks
1. As shown in the figure (1), there is no water in the sea at a height of 2m and an inclination angle of 30. What's the matter with you? /SPAN>。 _ _ _ _ _ _ meters.
Figure (1)
2. The cylindrical cup (as shown in the figure) for holding drinks is measured, and its inner bottom radius is 2.5㎝ and its height is 12㎝. Put the straw into the cup, and at least 4.6㎝ will be exposed outside the mouth of the cup. Let that fool do it.
3. Known: As shown in the figure, in △ABC, ∠C = 90? /SPAN>。 , point O is the intersection of three bisectors △ABC, OD ⊥ BC, OE ⊥ AC and of ⊥ AB, points D, E and F are vertical feet respectively, and BC = 8cm and CA = 6cm, then the distances from point O to three sides AB, AC and BC are equal to cm respectively.
There are two monkeys on a tree at the height of 10 meter. A monkey climbed down the tree and walked to the pond 20 meters away from the tree. The other monkey climbed to the top of the tree D and jumped directly into the pond A. The distance is calculated by a straight line. If two monkeys pass the same distance, then this tree is _ _ _ _ _ _ _ _ _ _ _.
5. The picture shows three steps, the length, width and height of each step are 20dm, 3dm,
2dm, A and B are the two opposite endpoints of this step. There is an ant at point a, thinking about B.
In order to eat delicious food, the shortest distance for ants to climb to point B along the step surface is _ _ _ _ _ _ _ _ _.
Second, multiple choice questions
1. Given an RT delta, the two sides are 3 and 4 respectively, and the square of the third side length is ().
A, 25 B, 14 C, 7 D, 7 or 25
2. The right-angle side length of Rt△ is 1 1, and the other two sides are natural numbers, so the circumference of RT △ is ().
A, 12 1 B, 120 C, 132 D, uncertain.
3. If the ratio of Rt△ two right angles is 5∶ 12, the ratio of the height on the hypotenuse to the hypotenuse is ().
a、60∶ 13 B、5∶ 12 C、 12∶ 13 D、60∶ 169
4. In the known Rt△ABC, ∠C=90 angstrom span >; A+b= 14cm and c= 10cm, then the area of Rt△ABC is ().
a、24cm2 B、36cm2 C、48cm2 D、60cm2
5. If the height on the bottom of an isosceles triangle is 8 and the circumference is 32, then the area of the triangle is ().
a、56 B、48 C、40 D、32
6. In the transformation of the old city, a city plans to plant turf in a triangular open space in the city, as shown in the figure, to beautify the environment. It is known that the price of this kind of turf is one yuan per square meter, so you need at least () to buy this kind of turf.
a 450 a b 225 a c 150 a d 300 a。
7. As we all know, as shown in the rectangle ABCD, AB=3cm and AD=9cm. If this rectangle is folded in half so that point B coincides with point D and the crease is EF, then the area of △ABE is () a, 6cm2 B, 8cm2 C, 10cm2 D, 12cm28 ... In △ABC, the area of △ Abe is. Height AD= 12, then the circumference of △ABC is a.42b.32c.42 or 32d.37 or 339. As shown in the figure, △ABC in a square grid, if the side length of a small square is 1, △ABC is
() (1) Right triangle
(b) acute triangle (c) obtuse triangle (d) None of the above answers are correct. 3. Calculate 1. As shown in the figure, A and B are two villages on the same side of the straight road L. The distance between the two villages is 300m and 500m respectively, and the distance between the two villages is D (d2=400000m2). Now a car will be built on the highway. 2. As shown in figure 1-3- 1 1, there is a plastic rectangular template ABCD, which is 10cm long and 4cm wide, which is big enough to hold in your hand.
PHF
The right-angled vertex P of the triangle falls on the edge of AD (not coincident with A and D), and move the triangle vertex P appropriately on AD: ① Can you make the two right-angled edges of your triangle pass through point B and point C respectively? Please make it clear this time, if you can.
The length of AP; If not, please explain why. (2) move the position of the triangle again, so that the vertex P of the triangle moves to AD, and the right angle side pH.
Always passing through point B, the extension line of another right-angle PF and DC intersects with point Q, and BC intersects with point E. Can CE =2cm? If possible, please find out the length of AP at this time; If not, please explain why.
Fourth, thinking training: 1. As shown in the figure, a rectangle with a length of 20cm and a width of 10cm is cut from the upper left corner of a rectangular steel plate with a length of 40cm and a width of 30cm. The remaining scraps are appropriately cut by the master, and then welded into a square workpiece with the same area as the original scraps and the shortest seam as possible. Please follow the above requirements. Two different splicing schemes are designed, and this scrap is properly divided into three or more pieces (in Figure 2 and Figure 3, the dotted line along the cutting and the square obtained after splicing are drawn respectively, leaving the traces of splicing). 2. kudzu vine is a tricky plant, and its waist is not hard. In order to compete for rain and sunshine, it often hovers around the trunk, and it also has a unique skill, that is, its rising route around the tree disk. If you read the above information, can you design a method to solve the following problems? If the circumference of a tree is 3 cm and it rises 4 cm in a circle, what is its crawling distance? If a tree is 8 cm in circumference and it crawls around 10 cm, how many centimeters will it rise when it crawls around? If you climb 10 times to reach the top of the tree, how many centimeters is the trunk high? 3. In △ABC, ∠ACB=90 angstrom? /SPAN>。 CD⊥AB in d, verification:.