There is a wonderful formula that allows you to count polygon area on the coordinate grid almost without errors. It's not even a formula, it's real theorem. At first glance, it may seem complicated. But it is enough to solve a couple of tasks - and you will understand how cool this feature is. So go ahead!
Let's start with a new definition:
A coordinate stack node is any point that lies at the intersection of the vertical and horizontal lines of this grid.
Designation:
In the first picture, the nodes are not marked at all. The second one has 4 nodes. Finally, in the third picture, all 16 nodes are marked.
What does this have to do with problem B5? The fact is that the vertices of the polygon in such problems always lie at the nodes of the grid. As a consequence, the following theorem works for them:
Theorem. Consider a polygon on a coordinate grid whose vertices lie at the nodes of this grid. Then the area of the polygon is:
where n is the number of nodes inside the given polygon, k is the number of nodes that lie on its boundary (boundary nodes).
As an example, consider an ordinary triangle on a coordinate grid and try to mark the internal and boundary nodes.
The first picture shows an ordinary triangle. On the second picture, its internal nodes are marked, the number of which is n = 10. On the third picture, the nodes lying on the border are marked, there are k = 6 of them in total.
Perhaps many readers do not understand how to count the numbers n and k. Start with internal nodes. Everything is obvious here: we paint over the triangle with a pencil and see how many nodes are shaded.
With boundary nodes, it's a little more complicated. polygon border - closed broken line, which intersects the coordinate grid at many points. The easiest way is to mark some "starting" point, and then go around the rest.
Boundary nodes will be only those points on the polyline at which they simultaneously intersect three lines:
- Actually, a broken line;
- Horizontal grid line;
- vertical line.
Let's see how it all works in real problems.
Task. Find the area of a triangle if the cell size is 1 x 1 cm:
First, let's mark the nodes that lie inside the triangle, as well as on its border:
It turns out that there is only one internal node: n = 1. There are six boundary nodes: three coincide with triangle vertices, and three more lie on the sides. Total k = 6.
Now we calculate the area using the formula:
That's all! Problem solved.
Task. Find the area of a quadrilateral depicted on checkered paper with a cell size of 1 cm by 1 cm. Give your answer in square centimeters.
Again, we mark the internal and boundary nodes. There are n = 2 internal nodes. Boundary nodes: k = 7, of which 4 are vertices of the quadrilateral, and 3 more lie on the sides.
It remains to substitute the numbers n and k in the area formula:
Pay attention to the last example. This problem was actually proposed at the diagnostic work in 2012. If you work according to the standard scheme, you will have to do a lot of additional constructions. And by the method of knots, everything is solved almost orally.
Important note on areas
But the formula is not everything. Let's rewrite the formula a little, bringing the terms on the right side to a common denominator. We get:
The numbers n and k are the number of nodes, they are always integers. So the whole numerator is also an integer. We divide it by 2, which implies an important fact:
The area is always expressed whole number or fraction. Moreover, at the end of the fraction there is always “five tenths”: 10.5; 17.5 etc.
Thus, the area in problem B5 is always expressed as an integer or a fraction of the form ***.5. If the answer is different, it means that a mistake has been made somewhere. Keep this in mind when you take the real exam in mathematics!
They are light, openwork, with a clear pattern of a mesh knot only when they are made of an even and smooth thread (no lint): this is a cord fishing line, linen threads, and highly twisted wool. By decreasing or increasing the distance between the nodes, the nets are woven around three-dimensional objects, exactly repeating their shape. But most of all, the monotonous unobtrusive texture of the grid is used as a background for the main pattern.
There are certain rules for weaving nets:
- the greater the gaps between the threads hung on the base, the more delicate the mesh;
- the number of threads hung on the warp must be a multiple of the number of threads in one knot;
- nodular and working threads of knots woven in a checkerboard pattern change places after each row, therefore they are consumed evenly;
- the number of rows is counted from the side edge of the grid (from top to bottom) along the extreme nodes, given that the nodes of the odd rows are located at the very edge, and the even rows are moved deeper;
- the number of hanging threads depends on the motif of the pattern (the motif is a certain number of knots rhythmically repeating in the pattern).
A grid of simple knots (Fig. 164, a).
The ends of the hanging threads for weaving the net should be 2 times longer than the height of the finished product. You can work from right to left and from left to right.
Fasten 4 threads 1 m long on the base at a distance of 2-3 cm from each other, folding them in half (8 threads turned out).
1st row - stepping back from the base by 1 cm, tie one left simple knot on each pair of threads (Fig. 164, b) and pin them with a pin to the half - it turned out 4 knots (Fig. 164, c).
2nd row - set aside the leftmost thread and on the next 2 (1 thread from the knots of the 1st row) tie a left simple knot, stepping back from the 1st row by 2-3 cm. Then on each next 2 threads weave knots, placing them on the same level with the 1st knot. The last rightmost thread remained free. 3 nodes were formed, which are staggered in relation to the nodes of the 1st row.
These knots are also pinned to the pillow.
3rd row - put into work 1 extreme free thread and 2-3 cm below the 2nd row, tie a simple knot on each pair of threads. 4 knots were formed (as in the 1st row).
4th row - weave as 2nd, trying to maintain equal distances between rows.
Figure 164, 165 and 166:
To prevent the mesh from slipping on the pillow, do not forget to pin the knots with pins - this will help to avoid skewed rows. Make sure that the loose threads along the edges of the mesh are the same size in all rows.
The mesh can be woven in a circle if you tie the ends of the warp. In this case, there are no free threads, since all the threads are involved in the weaving of each row.
A grid of intertwined simple knots (Fig. 165, a).
In this pattern, 2 simple knots are tied on every 2 threads, interlacing them with each other: first, the right simple knot is tied (not tight) on the left thread, then the right thread is pulled into the loop of the loose knot (Fig. 165, b) and the left simple knot is tied while tightening the right one. Both knots must intertwine symmetrically (Fig. 165, c) - such an interlacing is called a "love knot".
A grid of tie knots (Fig. 166).
The number of ends of the hanging threads is a multiple of 2. Their length should be 2 times the height of the net.
Grids from Armenian knots.
The calculation for a set of threads is the same as in the previous grid, if the knots are tied on 2 threads (Fig. 167, a). Another version of the pattern (Fig. 167, b): knots in odd rows are made on 3 threads, in even ones - on 2. The number of ends of the hanging threads in this pattern should be divided by 3. Their length is not the same (Fig. 168): short threads are equal height of the grid, long ones should be 2 times larger.
Scheme 167-168:
Nets of loop knots.
They are openwork or denser depending on the number of nodes in each link: the fewer nodes, the denser the mesh.
Grid of Single Right Nodes
(dense - Fig. 169). The number of ends of the hanging threads is a multiple of 2, their length should be 2 times the height of the grid.
Scheme 169-172:
For a mesh of triple one-sided knots
(all 3 right) - fig. 170, the number of ends of the hanging threads is a multiple of 4. The length of the odd threads should be equal to the height of the grid, the even ones - 2.5 times longer.
Openwork mesh from a combination of right and left loop knots
(Fig. 171) are made from threads that are 2.5 times longer than its height, their number should be divisible by 2.
Openwork grid with square cells
(Fig. 172) are woven from chains of "snakes", each of which consists of 4-6 loop knots. "Snakes" are connected in pairs with a double flat knot. The number of ends of the hanging threads for the net is a multiple of 4, their length must be 3 times the height of the net.
Grids from knots "tatting".
The knots in them are tied with a single thread (Fig. 173, a) and a double thread (Fig. 173, b). The length of the hanging threads in these two nets should be 3 times the height of the net. The number of ends for the first mesh must be divisible by 2, for the second - by 4.
Scheme 173-176:
Grid-meshka from knots "tatting" (fig. 174).
It is woven by alternating the right and left tatting knots: in odd rows, every 2 threads are tied with a left knot, in even rows, right knots are tied on the same knots, using working threads from the opposite knot of the previous row. In this case, the nodes are not arranged in a checkerboard pattern, but in vertical rows one below the other. The number of ends of the hanging threads is a multiple of 2 plus 1 for symmetry, the knotted threads must be equal to the length of the mesh, the working threads must be 2.5 times longer.
Grids from reps knots.
They are woven using diagonal and vertical rep knots.
When weaving a grid with large cells (Fig. 175, a), odd rows are performed on every 2 threads, tying them with diagonal knots from left to right (the right thread in the left hand is knotted, the left thread in the right hand is working). In even rows, the functions of the threads change: the knots of the previous row become working. Knots are performed from right to left.
If the mesh nodes are placed close to each other, then the weaving surface is finely booked (Fig. 175, b). This mesh is called tweed and boucle. The number of ends of the hanging threads must be divisible by 2, their length must be 4 times the height of the net.
Weaving a grid with triangular cells from vertical rep knots
(Fig. 176) is similar to weaving a horizontal brida from vertical rep knots(see Fig. 38, c).
The ends of the hanging threads should be the same length as the mesh, their number should be divisible by 2. Before work, all the threads are divided into 2 - nodular, and in the 1st row, an additional thread as a working thread is inserted into each pair of threads, one vertical rep node (right to left). In the next row, the knotted threads are redistributed in a checkerboard pattern and each pair of knotted threads is braided from left to right. The working thread for weaving can be cut as needed.
Grids of flat knots.
They are very popular in weaving. Some of them resemble lace, others, very dense, come with an intricate weave of threads.
A dense mesh of single flat knots is woven, tying the knots close to each other. The grid can be made more rare and embossed (Fig. 177) if each link is weaved from several knots (there should be enough of them for the warbler to turn on the edge).
The number of ends of the hanging threads for both nets is a multiple of 4. The length of the ends for the first net should be 3 times, and for the second net 5 times its height.
Scheme 177-180:
Grid of double flat knots - "checkerboard" (Fig. 178)
It has a clearer texture if the crossbars of all nodes are located on one side. The number of ends of the hanging threads is a multiple of 4. Their length should be 4 times the height of the net.
The structure of the double flat knot allows you to weave a net without hanging threads on the warp. Instead, pins are used: the threads are folded in half, pinned with a loop up to the pillow, and double flat knots are tied on every 4 of them (Fig. 179).
This technique is used when weaving clothes, shawls, i.e. when an elastic edge is needed.
For weaving a mesh of paired double flat knots on 6 threads
(Fig. 180) the ends of the hanging threads should be 3 times longer than the height of the grid, and their number should be divisible by 6. First, a double flat knot is tied on the middle 4 threads (2, 3, 4 and 5th) and the workers are taken aside threads (2nd and 5th), then tie the 2nd knot on the same knotted (3rd and 4th) free threads (1st and 6th). Starting and ending the next row, the first and last 3 threads are taken to the sides, the middle threads are redistributed into sixes.
By the same principle, a grid of double flat knots connected by 3 is woven (Fig. 181). The number of ends of the hanging threads is a multiple of 8, their length should be 2-3 times the height of the net.
A mesh of openwork gauges (Fig. 182) is performed on the same number of threads as the previous one, but the threads are redistributed after 2 groups of 3 knots are tied on the same nodular ones with an interval between groups of 0.5 cm.
Scheme 181-184:
In a grid of double flat knots - "polotnyanka"
(Fig. 183) - the nodes are not staggered, but one under the other. The number of ends of the hanging threads is a multiple of 4, their length should be 3 times the height of the net.
Grid-merezhka
(Fig. 184) is performed by crossing the working threads in each row. The length of the working threads is 4-6 times longer than the nodular threads, the length of the nodular threads is equal to the height of the mesh. The number of ends of the threads hung on the base should be divisible by 4. Make sure that the weave of the working threads is the same everywhere: right over left.
For mesh with large cells of double flat knots
(Fig. 185) the number of ends of the hanging threads is a multiple of 6 plus 4 threads for the symmetry of the pattern, the length of the knotted threads (2, 3, 5 and 6th) should be equal to the height of the grid, the working threads (1st and 4th) in 2 times longer.
Scheme 185-188
For a mesh of triple flat knots
(Fig. 186) the number of ends of the hanging threads is a multiple of 4, their length should be 4 times the height of the grid.
For a grid of flat complex knots
(Fig. 187) the number of ends of the hanging threads is a multiple of 6, their length should be 4 times the height of the grid.
For mesh with "flies"
(Fig. 188) the number of ends of the hanging threads should be divisible by 4, their length should be 3.5-4 times the height of the net. The "fly" is woven from 4 threads (1st and 4th - workers, 2nd and 3rd - knotted): first, the left single flat knot is tied, then the right simple knot is tied to 2 knotted ones (Fig. 189, a ) and under them - a double flat knot with a left crossbar (Fig. 189, b). If the "flies" are performed on thick threads, then instead of a double flat knot, a right single flat knot is tied (Fig. 189, c).
Scheme 189-191
Grid of Chinese knots (Fig. 190).
The number of ends of the hanging threads is a multiple of 2, their length should be 3 times the height of the mesh (see the knot tying technique in Fig. 139, a, b). If you hang the threads, alternating them in color, then the odd rows of the grid will be of two-color knots, and the even rows will be monochromatic (Fig. 191).
Grid of "Josephine" knots (Fig. 192).
It is better to perform it from bundles of threads or single, but thick threads. The number of ends of the hanging threads is a multiple of 2, their length must be 4 times the height of the net.
Scheme 192-193:
Grid of braids (Fig. 193).
It is rather complicated in execution, since when weaving it is difficult to maintain the same distance between the braids. The number of ends of the hanging threads is a multiple of 2, their length should be 4-5 times the height of the net.
Geometric mesh fragments.
The rhythm of the grid nodes is easily rebuilt into geometric shapes: hexagons (Fig. 194, a), triangles (Fig. 194, b), rhombuses (Fig. 194, c).
The technique of weaving them is very simple. For example, if a triangle is woven, then at the beginning and end of each new row, 2 extreme threads are not knitted, and then there will be 1 knot less in the row than in the previous one. So weave until only 1 knot remains. For weaving a triangle, the distribution of knots in rows is indicated by numbers, for example, 3, 2, 1. If a rhombus is performed, then they start with 1 knot: 1 double flat knot is tied on 4 middle threads, the threads are distributed under it by 2 and, adding 2 more free from the edges, tie under the 1st knot 2 double flat ones in a checkerboard pattern. In the following rows, they also attach
2 threads left and right, and as a result, the canvas expands after each row by 1 knot. Having completed the upper half of the rhombus, weave the lower one. Number of knots in rows: 1.2, 3.2, 1.
An example of another distribution of knots in rows 1, 2, 3, 4, 3, 2, 1 - according to this account, rhombuses are woven from Armenian (Fig. 195, a), diagonal rep (Fig. 195, b) and pioneer knots (Fig. 195, c) and from flat chains (Fig. 195, d).
Scheme 194 - 195
Grid nodes line up harmoniously in a horizontal
(Fig. 196, a) and vertical (Fig. 196, b) zigzag lines. It is more convenient to weave a horizontal one with separate corners, and then connect them with a knot using the threads of adjacent fragments. The corners are weaved like this: first, the upper central knot is tied, then the threads under it are divided equally (for example, 2 each) and sent 2 to the left, 2 to the right. Under the 1st knot on the left, a new knot is tied using 2 left threads from the 1st knot and 2 free ones on the left. Tie a knot on the right in the same way. Then weave another 1 knot on the right and left, using 2 free threads and 2 threads from the previous knot. So weave diagonally to the desired size of the corner with the top at the top. By connecting 2 corners with vertices directed in opposite directions, a rhombus is obtained, the middle of which is filled with free threads (Fig. 197, a). According to the same principle, a hexagon is woven (Fig. 197, b).
Scheme 196-199
A denser diagonal line of knots
(Fig. 198) is obtained if for each new node use 3 threads from the previous node (1 nodular, 2 workers) and 1 free.
Reception of connecting the grid with fragments of other patterns.
A monotonous web of mesh is used to create new patterns. On the grids, various woven fragments appear more embossed, convex and expressive. How are combinations of chains, knots and other elements woven into the grid? Get acquainted with these techniques using the example of a "checkerboard" grid in the middle with a common double flat knot (Fig. 199, a).
First, a sketch of a new pattern is drawn and the size and location of the knot are determined. As a rule, it is entered into a rhombus from free working threads (Fig. 199, b). The grid is woven to the top of the rhombus, then knots are not tied where the rhombus is marked: first 1 knot (the top of the rhombus), in the next row, 2 more knots in a checkerboard pattern, then 3 knots and so on until the desired width of the rhombus is obtained. After that, a common knot is tied. The more threads left for the knot, the more effective it is (when tying, do not skew it: the center of the knot should coincide with the center of the rhombus). Straighten the threads under the knot and weave the lower half of the rhombus, gradually including 2 extreme threads from the common knot into work.
