A few triangle questions, eager to !!!

Solution: Let the three sides of the triangle be a, b, c, and a b c

a+b+c=30，a+b＞c

10＜c＜15

c is an integer. c is 11, 12, 13, 14

When c is 14, there are 5 triangles, which are: 14, 13, 3; 14，12，4；14，11，5；14，10，6；14，9，7；

When c is 13, there are 4 triangles, which are: 13, 12, 5; 13，11，6；13，10，7；13，9，8；

When c is 12, there are 2 triangles, which are: 12, 11, 7; 12，10，8；

When c is 11, there is 1 triangle, which are: 11, 10, 9;

There are 12 triangles with unequal sides and integers

Solution: Let the waist length be a, then the bottom length is 2008-2a, the perimeter is 2008, 2a 2008, 2a 2008-2a, 2008-2a 2008, 502 a 1004, and the side length is an integer, so there are 501 kinds of isosceles triangles

2) Solution: Let this be an n-sided shape, and the degree of this inner angle is x degrees

Because (n-2)180°=2390°+x, so x=(n-2)180°-2390°=180°n-2750°,0 x 360°,0 180°n-2750° 360°,The solution:, n is odd, n=17, so the sum of the inner angles of the polygon is (17-2) 180°=2700°, that is, the degree of this inner angle is 2700°-2390°=310°

Solution: Let the number of external angles be x, according to the meaning of the question, get.

n-2) 180°+x=1350°, the solution is: x=1350°-180°N+360°=1710°-180°N, since 0 x 180°, i.e., 0 1710°-180°N 180°, the solution is obtained, so n=9

So the number of sides of the polygon is 9

The diagonal has n(n-3) 2=54 2=27

What does it mean to have a certificate that is both a certificate and a certificate that is a side length.

10cm

Because the distance from one point on the perpendicular bisector to both ends of the line segment is equal, BE=EA, the perimeter of the triangle BEC is BC+BE+EC, and the perimeter of the triangle ABC is AB+AC+BC

And because EC=EA (perpendicular bisector).

Equal substitution.

10cm

Draw a picture. C triangle BEC EC + CB + BE = 14cm, because according to the title AE = BE (the distance from the point on the perpendicular bisector to both sides of the line segment is equal), so the C triangle BEC EC + EA + BC = 14 AC + BC so ab 24-14 10cm

The sum of the two sides of the triangle should be greater than the third side.

ab+bc+ca=34

ab+1/2bc+ad=30

Because ab=ac

So the above equation becomes: 2ab+bc=34

Multiplying by 2 becomes: 2AB+BC+2AD=60, so AD=13

Center of gravity theorem: The three midlines of a triangle intersect at a point, and the distance from this point to the vertex is twice the distance from it to the midpoint of the opposite side. This point is called the center of gravity of the triangle.

Centroid theorem: The perpendicular bisector of the three sides of a triangle intersects at a point. This point is called the outer center of the triangle.

Perpendicular theorem: The three highs of a triangle intersect at one point. This point is called the vertical center of the triangle.

Inner theorem: The bisector of the three inside angles of a triangle intersects at one point. This point is called the triangle of the heart.

Centroid theorem: The bisector of one inner angle of a triangle and the bisector of the outer angle at the other two vertices intersect at one point. This point is called the paracentrium of the triangle. The triangle has three paracentricities.

The center of gravity, outer center, vertical center, inner center, and side center of the triangle are called the five hearts of the triangle. They are all important points of relevance for triangles.

The heart is the intersection of the bisector of the three corners of the triangle. The outer center is the intersection of the perpendicular bisector. The center of gravity is the intersection of the midlines of the three sides. The center of the vertical is the intersection of the three highs. That's all, I hope it can help you.

Let be rows 1 to nth from top to bottom, and let a(n) be the total number of triangles in the first n rows.

a(1) = 1

a(2) = a(1) +3 + 1

a(3) = a(2) +5 + 1 + 2

a(4) = a(3) +7 + 1 + 2 + 3

.a(n) = a(n-1) +2n-1 + 1+2+3+..n-1)

i.e. a(n) = a(n-1) +2n-1 + n(n-1) 2 = a(n-1) +n 2+3n) 2 - 1

So a(n) = a(1) +1 2[ 2 2 + 3 2 + n 2 + 3(2+3+4+..n)] n

n(n+1)(n+5)/6 - n

You can verify that, for example, the total number of triangles a(3) in the first 3 rows should be 1+3+5+1+1+2 = 13.

3*(3+1)(3+5) 6 -3 = 13, match.

No, there's also a missing handstand triangle.

Small regular triangle: 1+2+3+...11 66 small inverted triangle rows:

1+2+3+..10 55 Triangles of the size of the top two rows: 1+2+3+...

10 55 triangles in the top three rows: 1+2+3+...9 45 Triangles the size of the top four rows:

1+2+3+..8 36 Triangles in the top five rows: 1+2+3+...

7 28 Triangles in the top six rows: 1+2+3+...6 21 The top seven rows of triangles:

1+2+3+..5 15 The top eight rows of triangles: 1+2+3+...

4 10 The top nine rows of triangles: 1 + 2 + 3 6

The top ten rows of triangles: 1+2 3

The top eleven rows of triangles: 1 1

All the above triangles add up to the requirements: 66+55+55+45+36+28+21+15+10+6+3+1 341 Be polite Sorry, your answer is duplicated by others, please revise and submit.

**The method should be better

My answer is absolutely correct!!

When n is odd, the total number of triangles is (n+1)(2n 2+3n-1) 8

When n is even, the total number of triangles is n(n+2)(2n+1) 8

Here's how I understand it: the top layer is recorded as the first layer, and so on, we compare the second layer with the first layer, the second layer adds two triangles on the first layer, which is recorded as: 1+2, and according to this comparison, the third layer is:

1+2+2, the nth layer is noted: 1+2+2+2+....+2=1+2(n-1)=2n-1, so:

First layer: 2 1-1

Second layer: 2 2-1

Third layer: 2 3-1

Layer n: 2 n-1

Add them up and get: 2 (1+2+3+....)+n)-1 n=n If you count the 11 layers in this diagram, the answer is 121 small triangles, if your problem is to count the number of small triangles, this is how to understand!

Of course, if you really say how many triangles there are, the problem is a little more complicated!

1+2^2+……2^n

The first time is one! If you increase it to 2, you add 4, and if you increase it to 3, you add 8 more 4 times to 16!

Then there is when n=1, s=1;

n 1 s=2 (n+1)-3

n=11, s=4093

All triangles are counted, no matter how big or small!

Absolutely correct and the simplest general term.

Let the side length of the small triangle be 1, then the side length of any triangle in the figure is m, and there are a total of n triangles, n=11 in the figure

Let's start the summary.

All small triangles with a side length of 1 and upright triangles with a side length of 2 or more.

When n=1, (triangles of m=1) 1.

When n=2, there are 4 triangles (m=1) and 1 triangle (m=2).

At n=3, there are 9 triangles (m=1), 3 triangles (m=2), and 1 triangle (m=3).

At n = 4, there are 16 triangles (m=1), 6 triangles (m=2), 3 triangles (m=3), and 1 triangle (m=4).

At n = 5, there are 25 triangles (m=1), 10 triangles (m=2), 6 triangles (m=3), 3 triangles (m=4), and 1 triangle (m=5).

From this the general formula can be given.

n=n, n*n+n*(n-1) 2+(n-1)*(n-2) 2+(n-2)*(n-3) 2+(n-3)*(n-4) 2+......1

an=n^2+1/2*[(n-1)^2+(n-1)+(n-2)^2+(n-2)+…1^2+1]

n^2+1/2*[1/6*(n-1)*n*(2n-1)+n(n-1)/2]

n^2+n(n-1)/4+1/12*n(n-1)(2n-1)

1/6n^3+n^2-1/6n

An inverted triangle with a side length of 2 or more.

n=4，（m=2）=1

n=5，（m=2）=3

n=6，（m=2）=6，（m=3）=1

n=7，（m=2）=10，（m=3）=3

n=8，（m=2）=15，（m=3）=6，（m=4）=1

Hence we get bn=(n-2)(n-3) 2+(n-4)(n-5) 2+......1 (n is even).

1/12(n-3)(n-2)(2n-5)-1/12(n-2)(n-3)(n-4)+1/8(n-2)^2

1/12n^3-3/8n^2+5/12n

bn=（n-2)(n-3)/2+(n-4)(n-5)/2+……3 (n is an odd number).

1/12(n-1)(n-2)(n-3)+1/8(n-1)(n-3)

1/12n^3-3/8n^2+5/12n-1/8

Therefore, the odd and even numbers are combined to give bn=1 12n 3-3 8n 2+5 12n-1 16[1-(-1) n].

A total of an+bn triangles.

General Terms, General Terms, General Terms.

1/4n^3+5/8n^2+1/4n-1/16[1-(-1)^n]

When n=11, substitution yields 411

Look upwards of the triangle each line 1 2 3 4 5 6 7 8 9 10 11 the first row one, and each row below can form 10 triangles, that is, 10 + 1 = 11, the second row two, each with the following can form a triangle 2 9 itself two + 2 = 20 and so on The third row 3 8 + 3 = 3 9 = 27 The fourth row 4 8 = 32 The fifth row 5 7 = 35

Sixth line 6 6 = 36 Seventh row 7 5 = 35 Eighth row 8 4 = 32 Ninth row 9 3 = 27 Tenth row 10 2 = 20 The eleventh row itself 11 is 286 in total.

Look down on the triangle.

The bottom 10th row has 10 triangles.

With the ninth row can form 8.

With the eighth row can form 6.

With the seventh row you can form 4.

With the sixth row can form 2.

It is impossible to form a triangle with the fifth row.

The seventh row is 7 + 5 + 3 + 1 = 16

The sixth row is 6 + 4 + 2 = 12

The fifth row 5 + 3 + 1 = 9

Fourth row 4 + 2 = 6

The third row is 3+1=4

Second line 2, first line 1

30 + 25 + 20 + 16 + 12 + 9 + 6 + 4 + 2 + 1 = 125, so there are a total of 286 + 125 = 411 triangles.

I don't know if I'm doing the right thing or not, I think it's right. Do you see for yourself the same as you did?

Row 1 has 1 triangle s=2*1-1

There are 3 triangles in the second row: s=2*2-1

Row 3 has 5 triangles s=2*3-1

And so on the number of triangles in the nth row: s=2*n-1

Suppose the minimum unit of three sides is 1, first of all, the number of small triangles, it is easy to see that 11 squared, 121, and then there are 4 integrated triangles, the triangles are in different directions, as long as you determine how many vertices there are on the bottom edge, you can quickly calculate that it is obvious, the maximum base edge is 10 (11-1) vertices in the same direction (the vertices don't seem to have to be on top), so the triangle with the head up is 1 plus to 10, and then it is head down, and there are 8 (11-3) vertices on the bottom edge, which is 1 to 8, and then 10 Triangles up There are vertices with the same vertex direction as 11-2=9, and downwards are 11-5=6 that is, 1 is added to 8 In the same way, slowly push it up, you can also see that when the side length is 5, the triangle does not face down, and the result is 11 +1 plus to 10 + 1 to 9... 1 to 2 +1 plus 1 to 6... Add 1 to 2 +1 The answer is your own I don't know, you don't know if you don't have a blood difference series, so I won't talk about it.

It seems to be different from the answer, 11 side long, I greatly exceed 10 side long, I am 348, it seems that I was wrong, ashamed, ashamed.