Today, there is a pool of ten feet, born in the center, and one foot out of the water. Lead it ashore, and it will be suitable for landing. What is the water depth and the geometry of the water depth?
A: The water depth is one foot and two feet; This scab is one foot three feet long.
Technically, it is said: half a pool is self-multiplied, one foot is self-multiplied, and the rest is divided by twice the water, that is, the water depth is obtained. Add water, and it will grow.
Pythagorean theorem:
There is a ten-foot pool with reeds in the middle, which is one foot above the water. Drag the reeds to the shore, just in time to reach the shore. How deep is the water and how high is the reed?
A: The water depth is one foot two feet, and the reed is one foot three feet high.
Calculation method: Fold the side length of the pool into half a square, add the square of hydraulics, and subtract the sum of the former from the square of reed height to calculate the water depth, and add one foot to reed height.
That is, if the water depth is x and the reed height is (x+0. 1), there are:
2. Ask for help in translating the classical Chinese "Zhou Kuai Suan Jing". Zhou Kuai Suan Jing is one of the ten classic books of calculation. It was written in 1 century BC and was originally named Zhou Xie. It is the oldest astronomical work in China, which mainly expounds the theory of covering the sky and the method of seasonal 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.
One of China's earliest works on mathematics and astronomy. In ancient China, there were three kinds of astronomical theories according to different universe models, among which Gai Tian Shuo was one kind, and Zhouyi Tian Jing was the representative. The theory of this school holds that the sky is like a hat and the earth is like an overturned basin (the sky is like a hat and the earth is like an overturned basin).
According to textual research, Zhou pian Shu Jing was written in the Western Han Dynasty (BC 1 century). The block-printed edition of Southern Song Dynasty (Jiading 6th year, 12 13) is the earliest block-printed edition handed down at present, and it is collected in Shanghai Library. Many mathematicians in the past dynasties have annotated this book, the most famous of which is the annotation by Feng Chun and others in the Tang Dynasty. The classic of parallel week calculation has also spread to South Korea and Japan, where there are also many inscriptions.
Judging from the content of mathematics, the book mainly tells the method of learning mathematics, using Pythagorean theorem to calculate abstruse distance and complex fraction calculation.
There are mathematical contents in the book, such as the use of moments (tools for measuring right angles and drawing rectangles), Pythagorean theorem and its application in measurement, and the proportional theorem of corresponding sides of similar right triangles.
There are also the problems of square root and arithmetic progression, complicated fractional algorithm and Kaiping method, complicated fractional operation applied to the calculation of ancient "quarter calendar", complicated numerical calculation and the application of Pythagorean theorem.
The first chapter of the book describes the measurement method of Pythagorean theorem mentioned by Zhou Gong and Shang Gao in their questions and answers, and also gives a special case of "three strands, four strings and five".
Please punctuate the following two classical Chinese math problems and write the answers, 1 triangle geometry octagonal triangle three. First of all, what you entered may be wrong: triangle geometry * * octagon, triangle triangle, geometric geometry triangle+geometry = octagonal triangle = triangular money, so geometry = fifty cents. So the answer is: five cents is nine feet high, so it belongs to the book. ? Bamboo is broken, but not broken. The top (the end refers to the treetop) touches the ground and forms a right triangle with the remaining stumps planted in the soil and the ground. In which the oblique folding part is the hypotenuse C, the lower section of bamboo stands upright as the right-angled side B, and the bamboo on the ground is the other right-angled side A, where c+b=9 and a=3. Pythagorean theorem c set? -B? =3? =9，get (C+B) (C-B) = 9，C-B = 9/(C+B) = 9/9 = 1。 From the simultaneous equations of c+b=9 and c-b= 1, it is found that b=4 feet is an upright bamboo trunk.
4. Solve the Pythagorean Theorem problem (there is an answer), and all kinds of problems can be 1. In △ABC, a=(m+n) squared-1, b = 2m+2n, and c = (m+n) squared+1. Try to judge the shape of △ABC.
2. Given that AD is the height of △ABC, AD = the square of BDDC, is △ABC a right triangle? Explain why.
3. The ratio of the three internal angles of a triangle is 1:2:3, and its largest side is m, so what is its smallest side?
4. What is the area of an isosceles right triangle with the hypotenuse height m?
1. In △ABC, a=(m+n) squared-1, b = 2m+2n, and c = (m+n) squared+1. Try to judge the shape of △ABC.
Solution: 2 is a square and 4 is a power of 4.
a^2=(m+n)^4-2(m+n)^2+ 1
b^2=4(m+n)^2
c^2=(m+n)^4+2(m+n)^2+ 1
b^2=c^2-a^2
So the triangle is a right triangle.
2. Given that AD is the height of △ABC, AD = the square of BDDC, is △ABC a right triangle? Explain why.
Solution: △ABC is a right triangle, and 2 represents a square.
Derived from ad 2 = BD DC
AD/BD=DC/AD
Because AD is perpendicular to BC
So △ △ADB is similar to△△△ CDA.
So angle ABD= angle CAD angle BAD= angle ACD,
Because angle ADB= angle ADC=90 degrees
So angle ABD+ angle BAD=90 degrees.
So angle BAD+ angle CAD=90 degrees.
So a triangle is equivalent to a right triangle.
3. The ratio of the three internal angles of a triangle is 1:2:3, and its largest side is m, so what is its smallest side?
According to sine theorem, the minimum edge is 0.5m
4. What is the area of an isosceles right triangle with the hypotenuse height m?
Solution "
It is found that the bottom length is 2m,
S=0.5*m*2m=m*m
5. The ancient and modern meanings of the word "wolf" in classical Chinese, and the word "thigh ~ humerus" (also called left and right auxiliary)
In ancient China, there was a saying that an isosceles triangle formed the longer side of a right angle, such as Pythagorean theorem.
Imayoshi
1, human leg, that is, the part from hip to ankle, including thigh and calf. Especially the thigh, that is, the part from the hip to the knee: ~ bone. ~ humerus (also refers to people who can effectively assist).
2. A branch or part of something (a, capital share, such as "~", ~ "and ~"; B, a department in the organization; C others, such as "Chai ~" and "Ba ~ Wen").
3, quantifier (a, refers to a, such as "seven to the flood"; B, refers to the smell, such as "a ~ Xiang"; C, refers to strength, such as "twist into one strength"; D, batch, part, such as "a little enemy")
6. Pythagorean Theorem The mathematical thought in Pythagorean Theorem is the soul of solving mathematical problems, and the correct use of mathematical thought is also the key to successful problem solving.
When using Pythagorean theorem to solve problems, we should pay special attention to the application of mathematical thought. So what mathematical ideas are involved in solving Pythagorean theorem? Now give examples to illustrate the commonly used mathematical ideas in Pythagorean Theorem.
First, the example of equation thought 1 is shown in figure 1. In a rectangular ABCD, AD=6, AB=8, △ABD is folded in half along BD, and how long is the intersection of DC and F, CF? Solution: from the meaning of the question: △ Abd △ EBD, so ∠ABD=∠EBD. Because AB‖DC, so ∠ABD=∠BDC, so ∠EBD=∠BDC, so BF=DF.
Let CF=x, then BF = df = 8-X, in Rt△BCF, the solution is obtained, so 2. Example 2: The circumference of an isosceles triangle is 14cm, and one side is 4cm long. Find the height of the bottom.
Solution: (1) If 4cm is the waist length, then the bottom length is 6cm, then the height on the bottom. (2) If 4cm is the length of the bottom edge, the waist length is 5cm, and the height on the bottom edge.
So the height of the bottom edge. Third, the combination of numbers and shapes is shown in Figure 2. There are two monkeys on a tree at a height of 10 meters. One of them climbed down the tree and went straight to the pond 20 meters away from the tree. The other climbed to the top of the tree and went straight to the pond. If two monkeys pass the same distance, how tall is the tree? Solution: Let BD=x meters, from the meaning of the question, CD=(20-x) meters, AC= 10 meters.
In Rt△ACD, ∠ CAD = 90, so meters are needed to solve the equation. Then the height of this tree is () meters.
Answer: The height of this tree is () meters. Fourth, the change of ideas Example 4 is shown in Figure 3. The length of a cuboid is AB= 15cm, the width BC= 10cm, and the height BF=20cm. If an ant wants to climb from point A to point G along the surface of a cuboid, what is the shortest distance it needs to crawl? Solution: There are three situations: (1) As shown in Figure 4, the path AG is the shortest path for ants to crawl, at Rt△ACG, ∠ ACG = 90, AC=25cm, CG=20cm, then (2) As shown in Figure 5, the path AG is the shortest path for ants to crawl, at Rt△ABG. Then (3) As shown in Figure 6, the path AG is the shortest path for ants to crawl. In Rt△AFG, ∠ AFG = 90, AF=35cm, FG= 10cm, so the shortest path for ants to crawl is: Pythagorean theorem is a treasure of human beings and a wonderful work of mathematics. Pythagorean theorem contains rich mathematical ideas, and now it has been caught.