How to find the distance in the coordinate plane

In mathematics, both algebra and geometry set tasks to find 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 science, they teach how to use elementary tools (such as ruler, protractor, compass, triangle and others). Find the distance between points or lines with their help is not difficult. It is enough to attach the scale of divisions and record the answer. It is only necessary to know that the distance will be equal to the length of the straight line, which can be drawn between points, and in the case of parallel lines - the perpendicular between them.

The use of theorems and axioms of geometry

In the upper grades learn to measure the distance without the help of 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:

1. We connect points of a straight line, the length of which will be the distance between them.
2. We find the difference in the values of the coordinates of the points (k; p) of each axis: | k ₁ - k ₂ | = d ₁ and | p ₁ - p ₂ | = d ₂ (we take values modulo, since the distance can not be negative) .
3. After this, we construct the resulting numbers into squares and find their sum: d ₁ ² + d ₂ ²
4. The final step is the extraction of the square root of the resulting number. This will be the distance between the points: q = V (d ₁ ² + d ₂ ² ).

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

D = V (| k ₁ - k ₂ | ² + | p ₁ - p ₂ | ² )

If there is a question about how to find the distance from 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 implemented using the following formula:

Q = V (| k ₁ - k ₂ | ² + | p ₁ - p ₂ | ² + | e ₁ - e ₂ | ² )

Parallel straight lines

The 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 will be equal. In this case, the distance between parallel lines can be found by the formula:

D = | C ₁ - C ₂ | / V (A ² + B ² )

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