The master helps solve the math problems in the second year of junior high school, and explains it i

AD is perpendicular to EF.

Because AD is the angular bisector of ABC, de AB, DF AC, and perpendicular foot are E and F, respectively

So de df (the distance from the point on the angle bisector to both sides of the angle is equal) and ad ad (common edge).

So rt aed is all equal to rt afd(hl) so ae af (the corresponding sides of the congruent triangle are equal).

So AEF is an isosceles triangle.

And AD is the angle BAC angle bisector, that is, AG is the angle EAF angle bisector.

Therefore, AD is perpendicular to EF (the middle line of the bottom of the isosceles triangle, the angle bisector of the height and the apex angle are three-in-one).

Because AD is the angular bisector of ABC, DE AB, DF AC, so DE=DF, and because AE=AF, the triangle ADE is congruent with the triangle ADF, so AE=AF, and in the same way, it is proved that AEG is congruent with AFG, so the angle age is equal to the angle AGF, and because EGF is on the same line, it is 90°, so it is vertical.

According to the nature of the angular bisector, we know that de=df, in rt aed and rt afd, de df, ad=ad, i.e., they are congruent, so ae=af, angle eag=angular fag, in aeg and afg, ae=af,ag=ag, angular eag=angular fag, i.e., they are congruent, so the angle age=angle agf, and because they add 180°, one is 90°, i.e., ad, perpendicular ef.

1. From the original question, 3y-1=-(1-2x)=2x-1, so y=2x 3, can only get the relationship between x and y, but not the specific value.

The square root of , and the cubic root of 7, are not rational numbers, so they must be converted to rational numbers. The method is to the power of three numbers at the same time to the 6th power, and get , because 125>49>8, the square root of 5 >> the square root of 7 >> 2. The number with the root number is so compared.

3. Knowing v=, find s=6a 2, so, s square meters.

4. If the width is a meter, the length is 3a meter, and the length is 3a meter, s=3axa=3a 2=1200000000, so, a 2=40000000, a = 2000 root number 10, and because the root number is 10<5, so 2000 root number 10<10000. < 10 < can be taken as an approximation of the root number 10, so the width a = 6300 meters.

1. From the meaning of the title, it can be seen that 3y-1 and 1-2x are opposite numbers to each other, that is, 3y-1=-(1-2x).

If there are no special conditions.

There are countless solutions to this problem.

2. You don't need to count the cube root of 7.

The square root of 5 is, and the cube of the square root of 5 is.

The cube of the cube root of 7 is 7, and it is clear that 7 is less than.

Then the cube root of 7 is less than the cube root.

That is, the cubic root of 7 is less than the square root of 5.

3. The cube root of the edge length of the cube tank is meters.

Its surface area is: square meters.

4. Let its width be x, then its length is 3x

x×3×=120000000

3x squared = 120000000

x squared = 40000000

x=square root, I'll calculate, but it's not easy to say, just see if you can understand 1With the decimal point as the center, each two digits are divided into a section of 2Take the first section opening as an under-approximation, which is the first digit of the root.

3.Subtract the square of the first digit from the first stanza to get the difference and the second stanza to form the first remainder.

4.Multiply 20 by the first digit to try to divide the first remainder, and the test divisor is the second digit of the root.

5.Subtract the product of the second digit and the divisor from the first remainder, and the resulting difference and the next section spell out the second remainder.

6.Multiply 20 by the first two digits of the root, and try to divide the second remainder by the third digit of the root.

7.In this way, you can find the required number of digits.

Note: If the divisor is greater than or equal to 10 in the operation, 9 is taken

The cube root of 1:3y 1 and the cube root of 1 2x are opposite to each other, find x=? y=?

Answer: According to the question, 3y 1 (1 2x) 2x 1 can get the relationship between x and y, so x 3 2y, y 2 3 x.

2: 2, square root of 5, square root of 7, compare size. How do you calculate the cube root of 7?

Answer: There is no need to calculate the cube root of 7, just the power of the three numbers at the same time (the least common multiple of the three numbers open root exponents), this title is three numbers at the same time to the 6th power, and we get , because 125 49 8, so the square root of 5 The square root of 7 and the square root of 2.

3: A factory wants to weld a closed square water tank with a sheet of iron, his volume is cubic meters, and asks how much area of iron sheet is needed?

Answer: Let the side length of the cube be a, then the cubic meter of a is calculated as a meter; The side length of the iron sheet is 3a when the square water tank is closed, so the area of the iron sheet required is 3a 3a 9a a 9 square meters.

4: A development zone is a rectangle three times as long as it is wide, and its area is 120,000,000 square meters. Is there 10000 meters? If the required error is less than 100 meters, what is his width?

Answer: According to the topic, the width is set to a meter, then the length is 3a meters, and the area is 3a a 120000000, so a a 40000000, a 2000 root number 10, because the root number is 10 4, so the 2000 root number is 10 2000 4 8000, so the width is not 10000 meters. Whereas, it is desirable to take the average value with the root number 10 as an approximation, so that his width a 2000 m.

Another problem is that there is no way to get the exact cube root or square root of the number that cannot be divided like 2 and 7, but some such as the second question, what should I do if I need to compare the size? Note: Do not count as a cyclic decimal.

Answer: Like I answered the second question earlier, if you want to compare the size of several undivided numbers, you only need to count several numbers at the same time (the least common multiple of the opening exponent of several numbers) to the power, and it is easy to compare the size by unifying them into integers, and you don't need to count them as cyclic decimals.

1.Let it be: 3y-1=1-2x

So: 2x+3y-2=0

2.Multiply the root number 5 by the root number 3 7 by the 6th power at the same time, respectively, 125 and 49, so the former is 3Because its volume is cubic meters, its side length is meters. Therefore, its surface area is: 6* square meters) so the area of iron sheet is square meters.

4.Let the width be a, then there is: 3a 2 = 120000000, so a = 2000 meters ......

The cube root gives the side length of its square, and multiplies the square by 6 to get the square meter, so the area is square meters.

Do you need to ask someone else for this kind of math problem? You have to be valuable yourself! If you don't know, you can ask a teacher!

Solution: From 10x2-16xy+8y2+6x-4y+1=0, (9x2-12xy+4y2)+(6x-4y)+1+(4y2-4xy+x2)=0,3x-2y)2+2(3x-2y)+1+(2y-x)2=0,(3x-2y+1)2+(2y-x)2=0, 3x-2y+1=0,2y-x=0, x=,x-y=;

So the answer is: 4n 2 + 10n + 45 = x 2

4n^2+10n+45-x^2=0

The discriminant formula of the above equation =10 2-4*4*(45-x 2)=4*(4x 2-155).

n=[-5±√(4x^2-155)]/4

n is an integer, let 4x 2-155=y 2

4x^2-y^2=155

2x-y)*(2x+y)=1*155=5*31 If n has a maximum value, then y should take the maximum value, that is, 2x-y=1, y can obtain the maximum value, so the following system of equations:

2x-y=1...1)

2x+y=155...2)

2)-(1), got.

y=77x=39 is also an integer.

n=(-5±77)/4

Therefore n max = 18

Answer: The maximum value of the integer n = 18

The third question was not made.

The third method of undetermined coefficients:

Let the equation = (ax+by+c)*[1 a)x-(1 b)y+2 c] to find the coefficient.

Yes, the triangle bae is similar to the triangle cef, because the right-angled sides are all 1:2 long, and both have 1 right angle, which conforms to the corner edge similarity theorem, so bae= cef, and, bae+ bea=90°, so bea+ cef=90°, so aef=90°, so it is a right-angled triangle.

If cf=k, then ab=bc=cd=da=4k, be=ce=2k, cf=k, df=3k

According to the Pythagorean theorem, EF = root number 5k, ae = 2 times the root number 5k, af = 5k can be obtained to get the square of ae + the square of ef = the square of af.

So the triangle AEF is a right triangle.

Connect EF

There is a square ABCD

ab=bc=cd

ab/be=ce/cf=2/1

b=∠c=90°

ABE is similar to ECF.

So bae= cef

So CEF+ AEB=90°

So aef=90°

Suppose: Buy x large notebooks, then buy (5-x) small notebooks from the title:

6x+5(5-x)≤28

100x+60(5-x)≥340

Solution: 1 x 3

There are 3 schemes: 1 large book, 4 small books, a total of 340 pages, it costs 1 6 4 5 = 26 yuan.

2 large, 3 small, a total of 380 pages, it costs 2 6 3 5 = 27 yuan 3 large, 2 small, a total of 420 pages, it costs 3 6 2 5 = 28 yuan If in order to save money, then choose the least cost, but the most cost-effective is the plan.

The value of a large notebook is: 100 6=

The value of a small notebook is: 60 5 = 12

It is cost-effective to buy a large notebook, so you can buy 3 large and 1 small one, for a total of 360 pages.

Solution: Set up to buy x large notebooks.

It is derived from the title:

6x+5 (5-x) is less than or equal to 28

100x+60 (5-x) is greater than or equal to 340

Solving the inequality group yields: 3 is less than or equal to x less than or equal to 4

There are two purchase options:

1. 3 large notebooks and 2 small notebooks.

2. 4 large notebooks and 1 small notebook.