# How to find the distance in the coordinate plane

In mathematics both algebra and geometry are putthe problem of finding the distance to a point or a straight line from a given object. It is in completely different ways, the choice of which depends on the initial data. Consider how to find the distance between the given objects in different conditions.

*Use of measuring tools*

At the initial stage of mastering mathematical scienceteach how to use basic tools (such as ruler, protractor, compass, triangle and others). Finding the distance between points or lines with their help is not difficult. It is enough to attach a scale of divisions and write down the answer. It is only necessary to know that the distance will be equal to the length of a straight line, which can be drawn between points, and in the case of parallel lines - perpendicular between them.

*The use of theorems and axioms of geometry*

In the upper grades learn to measure distance withouthelp special tools or paper. For this, we need numerous theorems, axioms and their proofs. Often the problems of how to find the distance are reduced to the formation of a right triangle and the search for its sides. To solve such problems it is sufficient to know the Pythagorean theorem, the properties of triangles, and the ways of their transformation.

*Points on the coordinate plane*

If there are two points and their position is set on the coordinate axis, how to find the distance from one to the other? The solution will include several stages:

- We connect points of a straight line, the length of which will be the distance between them.
- We find the difference in the values of the coordinates of the points (k; p) of each axis: | k
_{1 }- to_{2}| = q_{1}and | p_{1 }- R_{2}| = q_{2}(we take values modulo, because the distance can not be negative). - After this, we construct the resulting numbers in a square and find their sum: q
_{1}^{2 }+ d_{2}^{2} - The final step is the extraction of the square root of the resulting number. This is the distance between the points: q = V (q
_{1}^{2 }+ d_{2}^{2}).

As a result, the entire solution is carried out according to one formula, where the distance is equal to the square root of the sum of the squares of the coordinate difference:

q = V (| k_{1 }- to_{2}| |^{2}+ | p_{1 }- R_{2}| |^{2})

If there is a question about how to find the distancefrom one point to another in three-dimensional space, the search for an answer to it will not be very different from the one given above. The solution will be carried out according to the following formula:

q = V (| k_{1 }- to_{2}| |^{2}+ | p_{1 }- R_{2}| |^{2}+ | e_{1 }- e_{2}| |^{2})

*Parallel straight lines*

A perpendicular drawn from any point,lying on one line, to the parallel, and is the distance. When solving problems in the plane, it is necessary to find the coordinates of any point of one of the lines. And then calculate the distance from it to the second straight line. For this, we reduce them to the general equation of a straight line of the form Ax + Bx + C = 0. It is known from the properties of parallel lines that their coefficients A and B are equal. In this case, the distance between parallel lines can be found from the formula:

q = | C_{1 }- C_{2}| / V (A^{2 }+ B^{2})

Thus, when answering the question of howfind the distance from the given object, it is necessary to be guided by the condition of the task and the tools to solve it. They can be both measuring devices, and theorems and formulas.

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